two version of R2R are here HEAD master

14 files changed, 5173 insertions, 0 deletions

diff --git a/.venv/lib/python3.12/site-packages/networkx/algorithms/tree/__init__.py b/.venv/lib/python3.12/site-packages/networkx/algorithms/tree/__init__.py
new file mode 100644
index 00000000..7120d4bc
--- /dev/null
+++ b/.venv/lib/python3.12/site-packages/networkx/algorithms/tree/__init__.py
@@ -0,0 +1,6 @@
+from .branchings import *
+from .coding import *
+from .mst import *
+from .recognition import *
+from .operations import *
+from .decomposition import *
diff --git a/.venv/lib/python3.12/site-packages/networkx/algorithms/tree/branchings.py b/.venv/lib/python3.12/site-packages/networkx/algorithms/tree/branchings.py
new file mode 100644
index 00000000..cc9c7cf1
--- /dev/null
+++ b/.venv/lib/python3.12/site-packages/networkx/algorithms/tree/branchings.py
@@ -0,0 +1,1042 @@
+"""
+Algorithms for finding optimum branchings and spanning arborescences.
+
+This implementation is based on:
+
+    J. Edmonds, Optimum branchings, J. Res. Natl. Bur. Standards 71B (1967),
+    233–240. URL: http://archive.org/details/jresv71Bn4p233
+
+"""
+
+# TODO: Implement method from Gabow, Galil, Spence and Tarjan:
+#
+# @article{
+#    year={1986},
+#    issn={0209-9683},
+#    journal={Combinatorica},
+#    volume={6},
+#    number={2},
+#    doi={10.1007/BF02579168},
+#    title={Efficient algorithms for finding minimum spanning trees in
+#        undirected and directed graphs},
+#    url={https://doi.org/10.1007/BF02579168},
+#    publisher={Springer-Verlag},
+#    keywords={68 B 15; 68 C 05},
+#    author={Gabow, Harold N. and Galil, Zvi and Spencer, Thomas and Tarjan,
+#        Robert E.},
+#    pages={109-122},
+#    language={English}
+# }
+import string
+from dataclasses import dataclass, field
+from operator import itemgetter
+from queue import PriorityQueue
+
+import networkx as nx
+from networkx.utils import py_random_state
+
+from .recognition import is_arborescence, is_branching
+
+__all__ = [
+    "branching_weight",
+    "greedy_branching",
+    "maximum_branching",
+    "minimum_branching",
+    "minimal_branching",
+    "maximum_spanning_arborescence",
+    "minimum_spanning_arborescence",
+    "ArborescenceIterator",
+]
+
+KINDS = {"max", "min"}
+
+STYLES = {
+    "branching": "branching",
+    "arborescence": "arborescence",
+    "spanning arborescence": "arborescence",
+}
+
+INF = float("inf")
+
+
+@py_random_state(1)
+def random_string(L=15, seed=None):
+    return "".join([seed.choice(string.ascii_letters) for n in range(L)])
+
+
+def _min_weight(weight):
+    return -weight
+
+
+def _max_weight(weight):
+    return weight
+
+
+@nx._dispatchable(edge_attrs={"attr": "default"})
+def branching_weight(G, attr="weight", default=1):
+    """
+    Returns the total weight of a branching.
+
+    You must access this function through the networkx.algorithms.tree module.
+
+    Parameters
+    ----------
+    G : DiGraph
+        The directed graph.
+    attr : str
+        The attribute to use as weights. If None, then each edge will be
+        treated equally with a weight of 1.
+    default : float
+        When `attr` is not None, then if an edge does not have that attribute,
+        `default` specifies what value it should take.
+
+    Returns
+    -------
+    weight: int or float
+        The total weight of the branching.
+
+    Examples
+    --------
+    >>> G = nx.DiGraph()
+    >>> G.add_weighted_edges_from([(0, 1, 2), (1, 2, 4), (2, 3, 3), (3, 4, 2)])
+    >>> nx.tree.branching_weight(G)
+    11
+
+    """
+    return sum(edge[2].get(attr, default) for edge in G.edges(data=True))
+
+
+@py_random_state(4)
+@nx._dispatchable(edge_attrs={"attr": "default"}, returns_graph=True)
+def greedy_branching(G, attr="weight", default=1, kind="max", seed=None):
+    """
+    Returns a branching obtained through a greedy algorithm.
+
+    This algorithm is wrong, and cannot give a proper optimal branching.
+    However, we include it for pedagogical reasons, as it can be helpful to
+    see what its outputs are.
+
+    The output is a branching, and possibly, a spanning arborescence. However,
+    it is not guaranteed to be optimal in either case.
+
+    Parameters
+    ----------
+    G : DiGraph
+        The directed graph to scan.
+    attr : str
+        The attribute to use as weights. If None, then each edge will be
+        treated equally with a weight of 1.
+    default : float
+        When `attr` is not None, then if an edge does not have that attribute,
+        `default` specifies what value it should take.
+    kind : str
+        The type of optimum to search for: 'min' or 'max' greedy branching.
+    seed : integer, random_state, or None (default)
+        Indicator of random number generation state.
+        See :ref:`Randomness<randomness>`.
+
+    Returns
+    -------
+    B : directed graph
+        The greedily obtained branching.
+
+    """
+    if kind not in KINDS:
+        raise nx.NetworkXException("Unknown value for `kind`.")
+
+    if kind == "min":
+        reverse = False
+    else:
+        reverse = True
+
+    if attr is None:
+        # Generate a random string the graph probably won't have.
+        attr = random_string(seed=seed)
+
+    edges = [(u, v, data.get(attr, default)) for (u, v, data) in G.edges(data=True)]
+
+    # We sort by weight, but also by nodes to normalize behavior across runs.
+    try:
+        edges.sort(key=itemgetter(2, 0, 1), reverse=reverse)
+    except TypeError:
+        # This will fail in Python 3.x if the nodes are of varying types.
+        # In that case, we use the arbitrary order.
+        edges.sort(key=itemgetter(2), reverse=reverse)
+
+    # The branching begins with a forest of no edges.
+    B = nx.DiGraph()
+    B.add_nodes_from(G)
+
+    # Now we add edges greedily so long we maintain the branching.
+    uf = nx.utils.UnionFind()
+    for i, (u, v, w) in enumerate(edges):
+        if uf[u] == uf[v]:
+            # Adding this edge would form a directed cycle.
+            continue
+        elif B.in_degree(v) == 1:
+            # The edge would increase the degree to be greater than one.
+            continue
+        else:
+            # If attr was None, then don't insert weights...
+            data = {}
+            if attr is not None:
+                data[attr] = w
+            B.add_edge(u, v, **data)
+            uf.union(u, v)
+
+    return B
+
+
+@nx._dispatchable(preserve_edge_attrs=True, returns_graph=True)
+def maximum_branching(
+    G,
+    attr="weight",
+    default=1,
+    preserve_attrs=False,
+    partition=None,
+):
+    #######################################
+    ### Data Structure Helper Functions ###
+    #######################################
+
+    def edmonds_add_edge(G, edge_index, u, v, key, **d):
+        """
+        Adds an edge to `G` while also updating the edge index.
+
+        This algorithm requires the use of an external dictionary to track
+        the edge keys since it is possible that the source or destination
+        node of an edge will be changed and the default key-handling
+        capabilities of the MultiDiGraph class do not account for this.
+
+        Parameters
+        ----------
+        G : MultiDiGraph
+            The graph to insert an edge into.
+        edge_index : dict
+            A mapping from integers to the edges of the graph.
+        u : node
+            The source node of the new edge.
+        v : node
+            The destination node of the new edge.
+        key : int
+            The key to use from `edge_index`.
+        d : keyword arguments, optional
+            Other attributes to store on the new edge.
+        """
+
+        if key in edge_index:
+            uu, vv, _ = edge_index[key]
+            if (u != uu) or (v != vv):
+                raise Exception(f"Key {key!r} is already in use.")
+
+        G.add_edge(u, v, key, **d)
+        edge_index[key] = (u, v, G.succ[u][v][key])
+
+    def edmonds_remove_node(G, edge_index, n):
+        """
+        Remove a node from the graph, updating the edge index to match.
+
+        Parameters
+        ----------
+        G : MultiDiGraph
+            The graph to remove an edge from.
+        edge_index : dict
+            A mapping from integers to the edges of the graph.
+        n : node
+            The node to remove from `G`.
+        """
+        keys = set()
+        for keydict in G.pred[n].values():
+            keys.update(keydict)
+        for keydict in G.succ[n].values():
+            keys.update(keydict)
+
+        for key in keys:
+            del edge_index[key]
+
+        G.remove_node(n)
+
+    #######################
+    ### Algorithm Setup ###
+    #######################
+
+    # Pick an attribute name that the original graph is unlikly to have
+    candidate_attr = "edmonds' secret candidate attribute"
+    new_node_base_name = "edmonds new node base name "
+
+    G_original = G
+    G = nx.MultiDiGraph()
+    G.__networkx_cache__ = None  # Disable caching
+
+    # A dict to reliably track mutations to the edges using the key of the edge.
+    G_edge_index = {}
+    # Each edge is given an arbitrary numerical key
+    for key, (u, v, data) in enumerate(G_original.edges(data=True)):
+        d = {attr: data.get(attr, default)}
+
+        if data.get(partition) is not None:
+            d[partition] = data.get(partition)
+
+        if preserve_attrs:
+            for d_k, d_v in data.items():
+                if d_k != attr:
+                    d[d_k] = d_v
+
+        edmonds_add_edge(G, G_edge_index, u, v, key, **d)
+
+    level = 0  # Stores the number of contracted nodes
+
+    # These are the buckets from the paper.
+    #
+    # In the paper, G^i are modified versions of the original graph.
+    # D^i and E^i are the nodes and edges of the maximal edges that are
+    # consistent with G^i. In this implementation, D^i and E^i are stored
+    # together as the graph B^i. We will have strictly more B^i then the
+    # paper will have.
+    #
+    # Note that the data in graphs and branchings are tuples with the graph as
+    # the first element and the edge index as the second.
+    B = nx.MultiDiGraph()
+    B_edge_index = {}
+    graphs = []  # G^i list
+    branchings = []  # B^i list
+    selected_nodes = set()  # D^i bucket
+    uf = nx.utils.UnionFind()
+
+    # A list of lists of edge indices. Each list is a circuit for graph G^i.
+    # Note the edge list is not required to be a circuit in G^0.
+    circuits = []
+
+    # Stores the index of the minimum edge in the circuit found in G^i and B^i.
+    # The ordering of the edges seems to preserver the weight ordering from
+    # G^0. So even if the circuit does not form a circuit in G^0, it is still
+    # true that the minimum edges in circuit G^0 (despite their weights being
+    # different)
+    minedge_circuit = []
+
+    ###########################
+    ### Algorithm Structure ###
+    ###########################
+
+    # Each step listed in the algorithm is an inner function. Thus, the overall
+    # loop structure is:
+    #
+    # while True:
+    #     step_I1()
+    #     if cycle detected:
+    #         step_I2()
+    #     elif every node of G is in D and E is a branching:
+    #         break
+
+    ##################################
+    ### Algorithm Helper Functions ###
+    ##################################
+
+    def edmonds_find_desired_edge(v):
+        """
+        Find the edge directed towards v with maximal weight.
+
+        If an edge partition exists in this graph, return the included
+        edge if it exists and never return any excluded edge.
+
+        Note: There can only be one included edge for each vertex otherwise
+        the edge partition is empty.
+
+        Parameters
+        ----------
+        v : node
+            The node to search for the maximal weight incoming edge.
+        """
+        edge = None
+        max_weight = -INF
+        for u, _, key, data in G.in_edges(v, data=True, keys=True):
+            # Skip excluded edges
+            if data.get(partition) == nx.EdgePartition.EXCLUDED:
+                continue
+
+            new_weight = data[attr]
+
+            # Return the included edge
+            if data.get(partition) == nx.EdgePartition.INCLUDED:
+                max_weight = new_weight
+                edge = (u, v, key, new_weight, data)
+                break
+
+            # Find the best open edge
+            if new_weight > max_weight:
+                max_weight = new_weight
+                edge = (u, v, key, new_weight, data)
+
+        return edge, max_weight
+
+    def edmonds_step_I2(v, desired_edge, level):
+        """
+        Perform step I2 from Edmonds' paper
+
+        First, check if the last step I1 created a cycle. If it did not, do nothing.
+        If it did, store the cycle for later reference and contract it.
+
+        Parameters
+        ----------
+        v : node
+            The current node to consider
+        desired_edge : edge
+            The minimum desired edge to remove from the cycle.
+        level : int
+            The current level, i.e. the number of cycles that have already been removed.
+        """
+        u = desired_edge[0]
+
+        Q_nodes = nx.shortest_path(B, v, u)
+        Q_edges = [
+            list(B[Q_nodes[i]][vv].keys())[0] for i, vv in enumerate(Q_nodes[1:])
+        ]
+        Q_edges.append(desired_edge[2])  # Add the new edge key to complete the circuit
+
+        # Get the edge in the circuit with the minimum weight.
+        # Also, save the incoming weights for each node.
+        minweight = INF
+        minedge = None
+        Q_incoming_weight = {}
+        for edge_key in Q_edges:
+            u, v, data = B_edge_index[edge_key]
+            w = data[attr]
+            # We cannot remove an included edge, even if it is the
+            # minimum edge in the circuit
+            Q_incoming_weight[v] = w
+            if data.get(partition) == nx.EdgePartition.INCLUDED:
+                continue
+            if w < minweight:
+                minweight = w
+                minedge = edge_key
+
+        circuits.append(Q_edges)
+        minedge_circuit.append(minedge)
+        graphs.append((G.copy(), G_edge_index.copy()))
+        branchings.append((B.copy(), B_edge_index.copy()))
+
+        # Mutate the graph to contract the circuit
+        new_node = new_node_base_name + str(level)
+        G.add_node(new_node)
+        new_edges = []
+        for u, v, key, data in G.edges(data=True, keys=True):
+            if u in Q_incoming_weight:
+                if v in Q_incoming_weight:
+                    # Circuit edge. For the moment do nothing,
+                    # eventually it will be removed.
+                    continue
+                else:
+                    # Outgoing edge from a node in the circuit.
+                    # Make it come from the new node instead
+                    dd = data.copy()
+                    new_edges.append((new_node, v, key, dd))
+            else:
+                if v in Q_incoming_weight:
+                    # Incoming edge to the circuit.
+                    # Update it's weight
+                    w = data[attr]
+                    w += minweight - Q_incoming_weight[v]
+                    dd = data.copy()
+                    dd[attr] = w
+                    new_edges.append((u, new_node, key, dd))
+                else:
+                    # Outside edge. No modification needed
+                    continue
+
+        for node in Q_nodes:
+            edmonds_remove_node(G, G_edge_index, node)
+            edmonds_remove_node(B, B_edge_index, node)
+
+        selected_nodes.difference_update(set(Q_nodes))
+
+        for u, v, key, data in new_edges:
+            edmonds_add_edge(G, G_edge_index, u, v, key, **data)
+            if candidate_attr in data:
+                del data[candidate_attr]
+                edmonds_add_edge(B, B_edge_index, u, v, key, **data)
+                uf.union(u, v)
+
+    def is_root(G, u, edgekeys):
+        """
+        Returns True if `u` is a root node in G.
+
+        Node `u` is a root node if its in-degree over the specified edges is zero.
+
+        Parameters
+        ----------
+        G : Graph
+            The current graph.
+        u : node
+            The node in `G` to check if it is a root.
+        edgekeys : iterable of edges
+            The edges for which to check if `u` is a root of.
+        """
+        if u not in G:
+            raise Exception(f"{u!r} not in G")
+
+        for v in G.pred[u]:
+            for edgekey in G.pred[u][v]:
+                if edgekey in edgekeys:
+                    return False, edgekey
+        else:
+            return True, None
+
+    nodes = iter(list(G.nodes))
+    while True:
+        try:
+            v = next(nodes)
+        except StopIteration:
+            # If there are no more new nodes to consider, then we should
+            # meet stopping condition (b) from the paper:
+            #   (b) every node of G^i is in D^i and E^i is a branching
+            assert len(G) == len(B)
+            if len(B):
+                assert is_branching(B)
+
+            graphs.append((G.copy(), G_edge_index.copy()))
+            branchings.append((B.copy(), B_edge_index.copy()))
+            circuits.append([])
+            minedge_circuit.append(None)
+
+            break
+        else:
+            #####################
+            ### BEGIN STEP I1 ###
+            #####################
+
+            # This is a very simple step, so I don't think it needs a method of it's own
+            if v in selected_nodes:
+                continue
+
+        selected_nodes.add(v)
+        B.add_node(v)
+        desired_edge, desired_edge_weight = edmonds_find_desired_edge(v)
+
+        # There might be no desired edge if all edges are excluded or
+        # v is the last node to be added to B, the ultimate root of the branching
+        if desired_edge is not None and desired_edge_weight > 0:
+            u = desired_edge[0]
+            # Flag adding the edge will create a circuit before merging the two
+            # connected components of u and v in B
+            circuit = uf[u] == uf[v]
+            dd = {attr: desired_edge_weight}
+            if desired_edge[4].get(partition) is not None:
+                dd[partition] = desired_edge[4].get(partition)
+
+            edmonds_add_edge(B, B_edge_index, u, v, desired_edge[2], **dd)
+            G[u][v][desired_edge[2]][candidate_attr] = True
+            uf.union(u, v)
+
+            ###################
+            ### END STEP I1 ###
+            ###################
+
+            #####################
+            ### BEGIN STEP I2 ###
+            #####################
+
+            if circuit:
+                edmonds_step_I2(v, desired_edge, level)
+                nodes = iter(list(G.nodes()))
+                level += 1
+
+            ###################
+            ### END STEP I2 ###
+            ###################
+
+    #####################
+    ### BEGIN STEP I3 ###
+    #####################
+
+    # Create a new graph of the same class as the input graph
+    H = G_original.__class__()
+
+    # Start with the branching edges in the last level.
+    edges = set(branchings[level][1])
+    while level > 0:
+        level -= 1
+
+        # The current level is i, and we start counting from 0.
+        #
+        # We need the node at level i+1 that results from merging a circuit
+        # at level i. basename_0 is the first merged node and this happens
+        # at level 1. That is basename_0 is a node at level 1 that results
+        # from merging a circuit at level 0.
+
+        merged_node = new_node_base_name + str(level)
+        circuit = circuits[level]
+        isroot, edgekey = is_root(graphs[level + 1][0], merged_node, edges)
+        edges.update(circuit)
+
+        if isroot:
+            minedge = minedge_circuit[level]
+            if minedge is None:
+                raise Exception
+
+            # Remove the edge in the cycle with minimum weight
+            edges.remove(minedge)
+        else:
+            # We have identified an edge at the next higher level that
+            # transitions into the merged node at this level. That edge
+            # transitions to some corresponding node at the current level.
+            #
+            # We want to remove an edge from the cycle that transitions
+            # into the corresponding node, otherwise the result would not
+            # be a branching.
+
+            G, G_edge_index = graphs[level]
+            target = G_edge_index[edgekey][1]
+            for edgekey in circuit:
+                u, v, data = G_edge_index[edgekey]
+                if v == target:
+                    break
+            else:
+                raise Exception("Couldn't find edge incoming to merged node.")
+
+            edges.remove(edgekey)
+
+    H.add_nodes_from(G_original)
+    for edgekey in edges:
+        u, v, d = graphs[0][1][edgekey]
+        dd = {attr: d[attr]}
+
+        if preserve_attrs:
+            for key, value in d.items():
+                if key not in [attr, candidate_attr]:
+                    dd[key] = value
+
+        H.add_edge(u, v, **dd)
+
+    ###################
+    ### END STEP I3 ###
+    ###################
+
+    return H
+
+
+@nx._dispatchable(preserve_edge_attrs=True, mutates_input=True, returns_graph=True)
+def minimum_branching(
+    G, attr="weight", default=1, preserve_attrs=False, partition=None
+):
+    for _, _, d in G.edges(data=True):
+        d[attr] = -d.get(attr, default)
+    nx._clear_cache(G)
+
+    B = maximum_branching(G, attr, default, preserve_attrs, partition)
+
+    for _, _, d in G.edges(data=True):
+        d[attr] = -d.get(attr, default)
+    nx._clear_cache(G)
+
+    for _, _, d in B.edges(data=True):
+        d[attr] = -d.get(attr, default)
+    nx._clear_cache(B)
+
+    return B
+
+
+@nx._dispatchable(preserve_edge_attrs=True, mutates_input=True, returns_graph=True)
+def minimal_branching(
+    G, /, *, attr="weight", default=1, preserve_attrs=False, partition=None
+):
+    """
+    Returns a minimal branching from `G`.
+
+    A minimal branching is a branching similar to a minimal arborescence but
+    without the requirement that the result is actually a spanning arborescence.
+    This allows minimal branchinges to be computed over graphs which may not
+    have arborescence (such as multiple components).
+
+    Parameters
+    ----------
+    G : (multi)digraph-like
+        The graph to be searched.
+    attr : str
+        The edge attribute used in determining optimality.
+    default : float
+        The value of the edge attribute used if an edge does not have
+        the attribute `attr`.
+    preserve_attrs : bool
+        If True, preserve the other attributes of the original graph (that are not
+        passed to `attr`)
+    partition : str
+        The key for the edge attribute containing the partition
+        data on the graph. Edges can be included, excluded or open using the
+        `EdgePartition` enum.
+
+    Returns
+    -------
+    B : (multi)digraph-like
+        A minimal branching.
+    """
+    max_weight = -INF
+    min_weight = INF
+    for _, _, w in G.edges(data=attr, default=default):
+        if w > max_weight:
+            max_weight = w
+        if w < min_weight:
+            min_weight = w
+
+    for _, _, d in G.edges(data=True):
+        # Transform the weights so that the minimum weight is larger than
+        # the difference between the max and min weights. This is important
+        # in order to prevent the edge weights from becoming negative during
+        # computation
+        d[attr] = max_weight + 1 + (max_weight - min_weight) - d.get(attr, default)
+    nx._clear_cache(G)
+
+    B = maximum_branching(G, attr, default, preserve_attrs, partition)
+
+    # Reverse the weight transformations
+    for _, _, d in G.edges(data=True):
+        d[attr] = max_weight + 1 + (max_weight - min_weight) - d.get(attr, default)
+    nx._clear_cache(G)
+
+    for _, _, d in B.edges(data=True):
+        d[attr] = max_weight + 1 + (max_weight - min_weight) - d.get(attr, default)
+    nx._clear_cache(B)
+
+    return B
+
+
+@nx._dispatchable(preserve_edge_attrs=True, mutates_input=True, returns_graph=True)
+def maximum_spanning_arborescence(
+    G, attr="weight", default=1, preserve_attrs=False, partition=None
+):
+    # In order to use the same algorithm is the maximum branching, we need to adjust
+    # the weights of the graph. The branching algorithm can choose to not include an
+    # edge if it doesn't help find a branching, mainly triggered by edges with negative
+    # weights.
+    #
+    # To prevent this from happening while trying to find a spanning arborescence, we
+    # just have to tweak the edge weights so that they are all positive and cannot
+    # become negative during the branching algorithm, find the maximum branching and
+    # then return them to their original values.
+
+    min_weight = INF
+    max_weight = -INF
+    for _, _, w in G.edges(data=attr, default=default):
+        if w < min_weight:
+            min_weight = w
+        if w > max_weight:
+            max_weight = w
+
+    for _, _, d in G.edges(data=True):
+        d[attr] = d.get(attr, default) - min_weight + 1 - (min_weight - max_weight)
+    nx._clear_cache(G)
+
+    B = maximum_branching(G, attr, default, preserve_attrs, partition)
+
+    for _, _, d in G.edges(data=True):
+        d[attr] = d.get(attr, default) + min_weight - 1 + (min_weight - max_weight)
+    nx._clear_cache(G)
+
+    for _, _, d in B.edges(data=True):
+        d[attr] = d.get(attr, default) + min_weight - 1 + (min_weight - max_weight)
+    nx._clear_cache(B)
+
+    if not is_arborescence(B):
+        raise nx.exception.NetworkXException("No maximum spanning arborescence in G.")
+
+    return B
+
+
+@nx._dispatchable(preserve_edge_attrs=True, mutates_input=True, returns_graph=True)
+def minimum_spanning_arborescence(
+    G, attr="weight", default=1, preserve_attrs=False, partition=None
+):
+    B = minimal_branching(
+        G,
+        attr=attr,
+        default=default,
+        preserve_attrs=preserve_attrs,
+        partition=partition,
+    )
+
+    if not is_arborescence(B):
+        raise nx.exception.NetworkXException("No minimum spanning arborescence in G.")
+
+    return B
+
+
+docstring_branching = """
+Returns a {kind} {style} from G.
+
+Parameters
+----------
+G : (multi)digraph-like
+    The graph to be searched.
+attr : str
+    The edge attribute used to in determining optimality.
+default : float
+    The value of the edge attribute used if an edge does not have
+    the attribute `attr`.
+preserve_attrs : bool
+    If True, preserve the other attributes of the original graph (that are not
+    passed to `attr`)
+partition : str
+    The key for the edge attribute containing the partition
+    data on the graph. Edges can be included, excluded or open using the
+    `EdgePartition` enum.
+
+Returns
+-------
+B : (multi)digraph-like
+    A {kind} {style}.
+"""
+
+docstring_arborescence = (
+    docstring_branching
+    + """
+Raises
+------
+NetworkXException
+    If the graph does not contain a {kind} {style}.
+
+"""
+)
+
+maximum_branching.__doc__ = docstring_branching.format(
+    kind="maximum", style="branching"
+)
+
+minimum_branching.__doc__ = (
+    docstring_branching.format(kind="minimum", style="branching")
+    + """
+See Also
+--------
+    minimal_branching
+"""
+)
+
+maximum_spanning_arborescence.__doc__ = docstring_arborescence.format(
+    kind="maximum", style="spanning arborescence"
+)
+
+minimum_spanning_arborescence.__doc__ = docstring_arborescence.format(
+    kind="minimum", style="spanning arborescence"
+)
+
+
+class ArborescenceIterator:
+    """
+    Iterate over all spanning arborescences of a graph in either increasing or
+    decreasing cost.
+
+    Notes
+    -----
+    This iterator uses the partition scheme from [1]_ (included edges,
+    excluded edges and open edges). It generates minimum spanning
+    arborescences using a modified Edmonds' Algorithm which respects the
+    partition of edges. For arborescences with the same weight, ties are
+    broken arbitrarily.
+
+    References
+    ----------
+    .. [1] G.K. Janssens, K. Sörensen, An algorithm to generate all spanning
+           trees in order of increasing cost, Pesquisa Operacional, 2005-08,
+           Vol. 25 (2), p. 219-229,
+           https://www.scielo.br/j/pope/a/XHswBwRwJyrfL88dmMwYNWp/?lang=en
+    """
+
+    @dataclass(order=True)
+    class Partition:
+        """
+        This dataclass represents a partition and stores a dict with the edge
+        data and the weight of the minimum spanning arborescence of the
+        partition dict.
+        """
+
+        mst_weight: float
+        partition_dict: dict = field(compare=False)
+
+        def __copy__(self):
+            return ArborescenceIterator.Partition(
+                self.mst_weight, self.partition_dict.copy()
+            )
+
+    def __init__(self, G, weight="weight", minimum=True, init_partition=None):
+        """
+        Initialize the iterator
+
+        Parameters
+        ----------
+        G : nx.DiGraph
+            The directed graph which we need to iterate trees over
+
+        weight : String, default = "weight"
+            The edge attribute used to store the weight of the edge
+
+        minimum : bool, default = True
+            Return the trees in increasing order while true and decreasing order
+            while false.
+
+        init_partition : tuple, default = None
+            In the case that certain edges have to be included or excluded from
+            the arborescences, `init_partition` should be in the form
+            `(included_edges, excluded_edges)` where each edges is a
+            `(u, v)`-tuple inside an iterable such as a list or set.
+
+        """
+        self.G = G.copy()
+        self.weight = weight
+        self.minimum = minimum
+        self.method = (
+            minimum_spanning_arborescence if minimum else maximum_spanning_arborescence
+        )
+        # Randomly create a key for an edge attribute to hold the partition data
+        self.partition_key = (
+            "ArborescenceIterators super secret partition attribute name"
+        )
+        if init_partition is not None:
+            partition_dict = {}
+            for e in init_partition[0]:
+                partition_dict[e] = nx.EdgePartition.INCLUDED
+            for e in init_partition[1]:
+                partition_dict[e] = nx.EdgePartition.EXCLUDED
+            self.init_partition = ArborescenceIterator.Partition(0, partition_dict)
+        else:
+            self.init_partition = None
+
+    def __iter__(self):
+        """
+        Returns
+        -------
+        ArborescenceIterator
+            The iterator object for this graph
+        """
+        self.partition_queue = PriorityQueue()
+        self._clear_partition(self.G)
+
+        # Write the initial partition if it exists.
+        if self.init_partition is not None:
+            self._write_partition(self.init_partition)
+
+        mst_weight = self.method(
+            self.G,
+            self.weight,
+            partition=self.partition_key,
+            preserve_attrs=True,
+        ).size(weight=self.weight)
+
+        self.partition_queue.put(
+            self.Partition(
+                mst_weight if self.minimum else -mst_weight,
+                (
+                    {}
+                    if self.init_partition is None
+                    else self.init_partition.partition_dict
+                ),
+            )
+        )
+
+        return self
+
+    def __next__(self):
+        """
+        Returns
+        -------
+        (multi)Graph
+            The spanning tree of next greatest weight, which ties broken
+            arbitrarily.
+        """
+        if self.partition_queue.empty():
+            del self.G, self.partition_queue
+            raise StopIteration
+
+        partition = self.partition_queue.get()
+        self._write_partition(partition)
+        next_arborescence = self.method(
+            self.G,
+            self.weight,
+            partition=self.partition_key,
+            preserve_attrs=True,
+        )
+        self._partition(partition, next_arborescence)
+
+        self._clear_partition(next_arborescence)
+        return next_arborescence
+
+    def _partition(self, partition, partition_arborescence):
+        """
+        Create new partitions based of the minimum spanning tree of the
+        current minimum partition.
+
+        Parameters
+        ----------
+        partition : Partition
+            The Partition instance used to generate the current minimum spanning
+            tree.
+        partition_arborescence : nx.Graph
+            The minimum spanning arborescence of the input partition.
+        """
+        # create two new partitions with the data from the input partition dict
+        p1 = self.Partition(0, partition.partition_dict.copy())
+        p2 = self.Partition(0, partition.partition_dict.copy())
+        for e in partition_arborescence.edges:
+            # determine if the edge was open or included
+            if e not in partition.partition_dict:
+                # This is an open edge
+                p1.partition_dict[e] = nx.EdgePartition.EXCLUDED
+                p2.partition_dict[e] = nx.EdgePartition.INCLUDED
+
+                self._write_partition(p1)
+                try:
+                    p1_mst = self.method(
+                        self.G,
+                        self.weight,
+                        partition=self.partition_key,
+                        preserve_attrs=True,
+                    )
+
+                    p1_mst_weight = p1_mst.size(weight=self.weight)
+                    p1.mst_weight = p1_mst_weight if self.minimum else -p1_mst_weight
+                    self.partition_queue.put(p1.__copy__())
+                except nx.NetworkXException:
+                    pass
+
+                p1.partition_dict = p2.partition_dict.copy()
+
+    def _write_partition(self, partition):
+        """
+        Writes the desired partition into the graph to calculate the minimum
+        spanning tree. Also, if one incoming edge is included, mark all others
+        as excluded so that if that vertex is merged during Edmonds' algorithm
+        we cannot still pick another of that vertex's included edges.
+
+        Parameters
+        ----------
+        partition : Partition
+            A Partition dataclass describing a partition on the edges of the
+            graph.
+        """
+        for u, v, d in self.G.edges(data=True):
+            if (u, v) in partition.partition_dict:
+                d[self.partition_key] = partition.partition_dict[(u, v)]
+            else:
+                d[self.partition_key] = nx.EdgePartition.OPEN
+        nx._clear_cache(self.G)
+
+        for n in self.G:
+            included_count = 0
+            excluded_count = 0
+            for u, v, d in self.G.in_edges(nbunch=n, data=True):
+                if d.get(self.partition_key) == nx.EdgePartition.INCLUDED:
+                    included_count += 1
+                elif d.get(self.partition_key) == nx.EdgePartition.EXCLUDED:
+                    excluded_count += 1
+            # Check that if there is an included edges, all other incoming ones
+            # are excluded. If not fix it!
+            if included_count == 1 and excluded_count != self.G.in_degree(n) - 1:
+                for u, v, d in self.G.in_edges(nbunch=n, data=True):
+                    if d.get(self.partition_key) != nx.EdgePartition.INCLUDED:
+                        d[self.partition_key] = nx.EdgePartition.EXCLUDED
+
+    def _clear_partition(self, G):
+        """
+        Removes partition data from the graph
+        """
+        for u, v, d in G.edges(data=True):
+            if self.partition_key in d:
+                del d[self.partition_key]
+        nx._clear_cache(self.G)
diff --git a/.venv/lib/python3.12/site-packages/networkx/algorithms/tree/coding.py b/.venv/lib/python3.12/site-packages/networkx/algorithms/tree/coding.py
new file mode 100644
index 00000000..f33089f7
--- /dev/null
+++ b/.venv/lib/python3.12/site-packages/networkx/algorithms/tree/coding.py
@@ -0,0 +1,413 @@
+"""Functions for encoding and decoding trees.
+
+Since a tree is a highly restricted form of graph, it can be represented
+concisely in several ways. This module includes functions for encoding
+and decoding trees in the form of nested tuples and Prüfer
+sequences. The former requires a rooted tree, whereas the latter can be
+applied to unrooted trees. Furthermore, there is a bijection from Prüfer
+sequences to labeled trees.
+
+"""
+
+from collections import Counter
+from itertools import chain
+
+import networkx as nx
+from networkx.utils import not_implemented_for
+
+__all__ = [
+    "from_nested_tuple",
+    "from_prufer_sequence",
+    "NotATree",
+    "to_nested_tuple",
+    "to_prufer_sequence",
+]
+
+
+class NotATree(nx.NetworkXException):
+    """Raised when a function expects a tree (that is, a connected
+    undirected graph with no cycles) but gets a non-tree graph as input
+    instead.
+
+    """
+
+
+@not_implemented_for("directed")
+@nx._dispatchable(graphs="T")
+def to_nested_tuple(T, root, canonical_form=False):
+    """Returns a nested tuple representation of the given tree.
+
+    The nested tuple representation of a tree is defined
+    recursively. The tree with one node and no edges is represented by
+    the empty tuple, ``()``. A tree with ``k`` subtrees is represented
+    by a tuple of length ``k`` in which each element is the nested tuple
+    representation of a subtree.
+
+    Parameters
+    ----------
+    T : NetworkX graph
+        An undirected graph object representing a tree.
+
+    root : node
+        The node in ``T`` to interpret as the root of the tree.
+
+    canonical_form : bool
+        If ``True``, each tuple is sorted so that the function returns
+        a canonical form for rooted trees. This means "lighter" subtrees
+        will appear as nested tuples before "heavier" subtrees. In this
+        way, each isomorphic rooted tree has the same nested tuple
+        representation.
+
+    Returns
+    -------
+    tuple
+        A nested tuple representation of the tree.
+
+    Notes
+    -----
+    This function is *not* the inverse of :func:`from_nested_tuple`; the
+    only guarantee is that the rooted trees are isomorphic.
+
+    See also
+    --------
+    from_nested_tuple
+    to_prufer_sequence
+
+    Examples
+    --------
+    The tree need not be a balanced binary tree::
+
+        >>> T = nx.Graph()
+        >>> T.add_edges_from([(0, 1), (0, 2), (0, 3)])
+        >>> T.add_edges_from([(1, 4), (1, 5)])
+        >>> T.add_edges_from([(3, 6), (3, 7)])
+        >>> root = 0
+        >>> nx.to_nested_tuple(T, root)
+        (((), ()), (), ((), ()))
+
+    Continuing the above example, if ``canonical_form`` is ``True``, the
+    nested tuples will be sorted::
+
+        >>> nx.to_nested_tuple(T, root, canonical_form=True)
+        ((), ((), ()), ((), ()))
+
+    Even the path graph can be interpreted as a tree::
+
+        >>> T = nx.path_graph(4)
+        >>> root = 0
+        >>> nx.to_nested_tuple(T, root)
+        ((((),),),)
+
+    """
+
+    def _make_tuple(T, root, _parent):
+        """Recursively compute the nested tuple representation of the
+        given rooted tree.
+
+        ``_parent`` is the parent node of ``root`` in the supertree in
+        which ``T`` is a subtree, or ``None`` if ``root`` is the root of
+        the supertree. This argument is used to determine which
+        neighbors of ``root`` are children and which is the parent.
+
+        """
+        # Get the neighbors of `root` that are not the parent node. We
+        # are guaranteed that `root` is always in `T` by construction.
+        children = set(T[root]) - {_parent}
+        if len(children) == 0:
+            return ()
+        nested = (_make_tuple(T, v, root) for v in children)
+        if canonical_form:
+            nested = sorted(nested)
+        return tuple(nested)
+
+    # Do some sanity checks on the input.
+    if not nx.is_tree(T):
+        raise nx.NotATree("provided graph is not a tree")
+    if root not in T:
+        raise nx.NodeNotFound(f"Graph {T} contains no node {root}")
+
+    return _make_tuple(T, root, None)
+
+
+@nx._dispatchable(graphs=None, returns_graph=True)
+def from_nested_tuple(sequence, sensible_relabeling=False):
+    """Returns the rooted tree corresponding to the given nested tuple.
+
+    The nested tuple representation of a tree is defined
+    recursively. The tree with one node and no edges is represented by
+    the empty tuple, ``()``. A tree with ``k`` subtrees is represented
+    by a tuple of length ``k`` in which each element is the nested tuple
+    representation of a subtree.
+
+    Parameters
+    ----------
+    sequence : tuple
+        A nested tuple representing a rooted tree.
+
+    sensible_relabeling : bool
+        Whether to relabel the nodes of the tree so that nodes are
+        labeled in increasing order according to their breadth-first
+        search order from the root node.
+
+    Returns
+    -------
+    NetworkX graph
+        The tree corresponding to the given nested tuple, whose root
+        node is node 0. If ``sensible_labeling`` is ``True``, nodes will
+        be labeled in breadth-first search order starting from the root
+        node.
+
+    Notes
+    -----
+    This function is *not* the inverse of :func:`to_nested_tuple`; the
+    only guarantee is that the rooted trees are isomorphic.
+
+    See also
+    --------
+    to_nested_tuple
+    from_prufer_sequence
+
+    Examples
+    --------
+    Sensible relabeling ensures that the nodes are labeled from the root
+    starting at 0::
+
+        >>> balanced = (((), ()), ((), ()))
+        >>> T = nx.from_nested_tuple(balanced, sensible_relabeling=True)
+        >>> edges = [(0, 1), (0, 2), (1, 3), (1, 4), (2, 5), (2, 6)]
+        >>> all((u, v) in T.edges() or (v, u) in T.edges() for (u, v) in edges)
+        True
+
+    """
+
+    def _make_tree(sequence):
+        """Recursively creates a tree from the given sequence of nested
+        tuples.
+
+        This function employs the :func:`~networkx.tree.join` function
+        to recursively join subtrees into a larger tree.
+
+        """
+        # The empty sequence represents the empty tree, which is the
+        # (unique) graph with a single node. We mark the single node
+        # with an attribute that indicates that it is the root of the
+        # graph.
+        if len(sequence) == 0:
+            return nx.empty_graph(1)
+        # For a nonempty sequence, get the subtrees for each child
+        # sequence and join all the subtrees at their roots. After
+        # joining the subtrees, the root is node 0.
+        return nx.tree.join_trees([(_make_tree(child), 0) for child in sequence])
+
+    # Make the tree and remove the `is_root` node attribute added by the
+    # helper function.
+    T = _make_tree(sequence)
+    if sensible_relabeling:
+        # Relabel the nodes according to their breadth-first search
+        # order, starting from the root node (that is, the node 0).
+        bfs_nodes = chain([0], (v for u, v in nx.bfs_edges(T, 0)))
+        labels = {v: i for i, v in enumerate(bfs_nodes)}
+        # We would like to use `copy=False`, but `relabel_nodes` doesn't
+        # allow a relabel mapping that can't be topologically sorted.
+        T = nx.relabel_nodes(T, labels)
+    return T
+
+
+@not_implemented_for("directed")
+@nx._dispatchable(graphs="T")
+def to_prufer_sequence(T):
+    r"""Returns the Prüfer sequence of the given tree.
+
+    A *Prüfer sequence* is a list of *n* - 2 numbers between 0 and
+    *n* - 1, inclusive. The tree corresponding to a given Prüfer
+    sequence can be recovered by repeatedly joining a node in the
+    sequence with a node with the smallest potential degree according to
+    the sequence.
+
+    Parameters
+    ----------
+    T : NetworkX graph
+        An undirected graph object representing a tree.
+
+    Returns
+    -------
+    list
+        The Prüfer sequence of the given tree.
+
+    Raises
+    ------
+    NetworkXPointlessConcept
+        If the number of nodes in `T` is less than two.
+
+    NotATree
+        If `T` is not a tree.
+
+    KeyError
+        If the set of nodes in `T` is not {0, …, *n* - 1}.
+
+    Notes
+    -----
+    There is a bijection from labeled trees to Prüfer sequences. This
+    function is the inverse of the :func:`from_prufer_sequence`
+    function.
+
+    Sometimes Prüfer sequences use nodes labeled from 1 to *n* instead
+    of from 0 to *n* - 1. This function requires nodes to be labeled in
+    the latter form. You can use :func:`~networkx.relabel_nodes` to
+    relabel the nodes of your tree to the appropriate format.
+
+    This implementation is from [1]_ and has a running time of
+    $O(n)$.
+
+    See also
+    --------
+    to_nested_tuple
+    from_prufer_sequence
+
+    References
+    ----------
+    .. [1] Wang, Xiaodong, Lei Wang, and Yingjie Wu.
+           "An optimal algorithm for Prufer codes."
+           *Journal of Software Engineering and Applications* 2.02 (2009): 111.
+           <https://doi.org/10.4236/jsea.2009.22016>
+
+    Examples
+    --------
+    There is a bijection between Prüfer sequences and labeled trees, so
+    this function is the inverse of the :func:`from_prufer_sequence`
+    function:
+
+    >>> edges = [(0, 3), (1, 3), (2, 3), (3, 4), (4, 5)]
+    >>> tree = nx.Graph(edges)
+    >>> sequence = nx.to_prufer_sequence(tree)
+    >>> sequence
+    [3, 3, 3, 4]
+    >>> tree2 = nx.from_prufer_sequence(sequence)
+    >>> list(tree2.edges()) == edges
+    True
+
+    """
+    # Perform some sanity checks on the input.
+    n = len(T)
+    if n < 2:
+        msg = "Prüfer sequence undefined for trees with fewer than two nodes"
+        raise nx.NetworkXPointlessConcept(msg)
+    if not nx.is_tree(T):
+        raise nx.NotATree("provided graph is not a tree")
+    if set(T) != set(range(n)):
+        raise KeyError("tree must have node labels {0, ..., n - 1}")
+
+    degree = dict(T.degree())
+
+    def parents(u):
+        return next(v for v in T[u] if degree[v] > 1)
+
+    index = u = next(k for k in range(n) if degree[k] == 1)
+    result = []
+    for i in range(n - 2):
+        v = parents(u)
+        result.append(v)
+        degree[v] -= 1
+        if v < index and degree[v] == 1:
+            u = v
+        else:
+            index = u = next(k for k in range(index + 1, n) if degree[k] == 1)
+    return result
+
+
+@nx._dispatchable(graphs=None, returns_graph=True)
+def from_prufer_sequence(sequence):
+    r"""Returns the tree corresponding to the given Prüfer sequence.
+
+    A *Prüfer sequence* is a list of *n* - 2 numbers between 0 and
+    *n* - 1, inclusive. The tree corresponding to a given Prüfer
+    sequence can be recovered by repeatedly joining a node in the
+    sequence with a node with the smallest potential degree according to
+    the sequence.
+
+    Parameters
+    ----------
+    sequence : list
+        A Prüfer sequence, which is a list of *n* - 2 integers between
+        zero and *n* - 1, inclusive.
+
+    Returns
+    -------
+    NetworkX graph
+        The tree corresponding to the given Prüfer sequence.
+
+    Raises
+    ------
+    NetworkXError
+        If the Prüfer sequence is not valid.
+
+    Notes
+    -----
+    There is a bijection from labeled trees to Prüfer sequences. This
+    function is the inverse of the :func:`from_prufer_sequence` function.
+
+    Sometimes Prüfer sequences use nodes labeled from 1 to *n* instead
+    of from 0 to *n* - 1. This function requires nodes to be labeled in
+    the latter form. You can use :func:`networkx.relabel_nodes` to
+    relabel the nodes of your tree to the appropriate format.
+
+    This implementation is from [1]_ and has a running time of
+    $O(n)$.
+
+    References
+    ----------
+    .. [1] Wang, Xiaodong, Lei Wang, and Yingjie Wu.
+           "An optimal algorithm for Prufer codes."
+           *Journal of Software Engineering and Applications* 2.02 (2009): 111.
+           <https://doi.org/10.4236/jsea.2009.22016>
+
+    See also
+    --------
+    from_nested_tuple
+    to_prufer_sequence
+
+    Examples
+    --------
+    There is a bijection between Prüfer sequences and labeled trees, so
+    this function is the inverse of the :func:`to_prufer_sequence`
+    function:
+
+    >>> edges = [(0, 3), (1, 3), (2, 3), (3, 4), (4, 5)]
+    >>> tree = nx.Graph(edges)
+    >>> sequence = nx.to_prufer_sequence(tree)
+    >>> sequence
+    [3, 3, 3, 4]
+    >>> tree2 = nx.from_prufer_sequence(sequence)
+    >>> list(tree2.edges()) == edges
+    True
+
+    """
+    n = len(sequence) + 2
+    # `degree` stores the remaining degree (plus one) for each node. The
+    # degree of a node in the decoded tree is one more than the number
+    # of times it appears in the code.
+    degree = Counter(chain(sequence, range(n)))
+    T = nx.empty_graph(n)
+    # `not_orphaned` is the set of nodes that have a parent in the
+    # tree. After the loop, there should be exactly two nodes that are
+    # not in this set.
+    not_orphaned = set()
+    index = u = next(k for k in range(n) if degree[k] == 1)
+    for v in sequence:
+        # check the validity of the prufer sequence
+        if v < 0 or v > n - 1:
+            raise nx.NetworkXError(
+                f"Invalid Prufer sequence: Values must be between 0 and {n-1}, got {v}"
+            )
+        T.add_edge(u, v)
+        not_orphaned.add(u)
+        degree[v] -= 1
+        if v < index and degree[v] == 1:
+            u = v
+        else:
+            index = u = next(k for k in range(index + 1, n) if degree[k] == 1)
+    # At this point, there must be exactly two orphaned nodes; join them.
+    orphans = set(T) - not_orphaned
+    u, v = orphans
+    T.add_edge(u, v)
+    return T
diff --git a/.venv/lib/python3.12/site-packages/networkx/algorithms/tree/decomposition.py b/.venv/lib/python3.12/site-packages/networkx/algorithms/tree/decomposition.py
new file mode 100644
index 00000000..c8b8f247
--- /dev/null
+++ b/.venv/lib/python3.12/site-packages/networkx/algorithms/tree/decomposition.py
@@ -0,0 +1,88 @@
+r"""Function for computing a junction tree of a graph."""
+
+from itertools import combinations
+
+import networkx as nx
+from networkx.algorithms import chordal_graph_cliques, complete_to_chordal_graph, moral
+from networkx.utils import not_implemented_for
+
+__all__ = ["junction_tree"]
+
+
+@not_implemented_for("multigraph")
+@nx._dispatchable(returns_graph=True)
+def junction_tree(G):
+    r"""Returns a junction tree of a given graph.
+
+    A junction tree (or clique tree) is constructed from a (un)directed graph G.
+    The tree is constructed based on a moralized and triangulated version of G.
+    The tree's nodes consist of maximal cliques and sepsets of the revised graph.
+    The sepset of two cliques is the intersection of the nodes of these cliques,
+    e.g. the sepset of (A,B,C) and (A,C,E,F) is (A,C). These nodes are often called
+    "variables" in this literature. The tree is bipartite with each sepset
+    connected to its two cliques.
+
+    Junction Trees are not unique as the order of clique consideration determines
+    which sepsets are included.
+
+    The junction tree algorithm consists of five steps [1]_:
+
+    1. Moralize the graph
+    2. Triangulate the graph
+    3. Find maximal cliques
+    4. Build the tree from cliques, connecting cliques with shared
+       nodes, set edge-weight to number of shared variables
+    5. Find maximum spanning tree
+
+
+    Parameters
+    ----------
+    G : networkx.Graph
+        Directed or undirected graph.
+
+    Returns
+    -------
+    junction_tree : networkx.Graph
+        The corresponding junction tree of `G`.
+
+    Raises
+    ------
+    NetworkXNotImplemented
+        Raised if `G` is an instance of `MultiGraph` or `MultiDiGraph`.
+
+    References
+    ----------
+    .. [1] Junction tree algorithm:
+       https://en.wikipedia.org/wiki/Junction_tree_algorithm
+
+    .. [2] Finn V. Jensen and Frank Jensen. 1994. Optimal
+       junction trees. In Proceedings of the Tenth international
+       conference on Uncertainty in artificial intelligence (UAI’94).
+       Morgan Kaufmann Publishers Inc., San Francisco, CA, USA, 360–366.
+    """
+
+    clique_graph = nx.Graph()
+
+    if G.is_directed():
+        G = moral.moral_graph(G)
+    chordal_graph, _ = complete_to_chordal_graph(G)
+
+    cliques = [tuple(sorted(i)) for i in chordal_graph_cliques(chordal_graph)]
+    clique_graph.add_nodes_from(cliques, type="clique")
+
+    for edge in combinations(cliques, 2):
+        set_edge_0 = set(edge[0])
+        set_edge_1 = set(edge[1])
+        if not set_edge_0.isdisjoint(set_edge_1):
+            sepset = tuple(sorted(set_edge_0.intersection(set_edge_1)))
+            clique_graph.add_edge(edge[0], edge[1], weight=len(sepset), sepset=sepset)
+
+    junction_tree = nx.maximum_spanning_tree(clique_graph)
+
+    for edge in list(junction_tree.edges(data=True)):
+        junction_tree.add_node(edge[2]["sepset"], type="sepset")
+        junction_tree.add_edge(edge[0], edge[2]["sepset"])
+        junction_tree.add_edge(edge[1], edge[2]["sepset"])
+        junction_tree.remove_edge(edge[0], edge[1])
+
+    return junction_tree
diff --git a/.venv/lib/python3.12/site-packages/networkx/algorithms/tree/mst.py b/.venv/lib/python3.12/site-packages/networkx/algorithms/tree/mst.py
new file mode 100644
index 00000000..554613b8
--- /dev/null
+++ b/.venv/lib/python3.12/site-packages/networkx/algorithms/tree/mst.py
@@ -0,0 +1,1284 @@
+"""
+Algorithms for calculating min/max spanning trees/forests.
+
+"""
+
+from dataclasses import dataclass, field
+from enum import Enum
+from heapq import heappop, heappush
+from itertools import count
+from math import isnan
+from operator import itemgetter
+from queue import PriorityQueue
+
+import networkx as nx
+from networkx.utils import UnionFind, not_implemented_for, py_random_state
+
+__all__ = [
+    "minimum_spanning_edges",
+    "maximum_spanning_edges",
+    "minimum_spanning_tree",
+    "maximum_spanning_tree",
+    "number_of_spanning_trees",
+    "random_spanning_tree",
+    "partition_spanning_tree",
+    "EdgePartition",
+    "SpanningTreeIterator",
+]
+
+
+class EdgePartition(Enum):
+    """
+    An enum to store the state of an edge partition. The enum is written to the
+    edges of a graph before being pasted to `kruskal_mst_edges`. Options are:
+
+    - EdgePartition.OPEN
+    - EdgePartition.INCLUDED
+    - EdgePartition.EXCLUDED
+    """
+
+    OPEN = 0
+    INCLUDED = 1
+    EXCLUDED = 2
+
+
+@not_implemented_for("multigraph")
+@nx._dispatchable(edge_attrs="weight", preserve_edge_attrs="data")
+def boruvka_mst_edges(
+    G, minimum=True, weight="weight", keys=False, data=True, ignore_nan=False
+):
+    """Iterate over edges of a Borůvka's algorithm min/max spanning tree.
+
+    Parameters
+    ----------
+    G : NetworkX Graph
+        The edges of `G` must have distinct weights,
+        otherwise the edges may not form a tree.
+
+    minimum : bool (default: True)
+        Find the minimum (True) or maximum (False) spanning tree.
+
+    weight : string (default: 'weight')
+        The name of the edge attribute holding the edge weights.
+
+    keys : bool (default: True)
+        This argument is ignored since this function is not
+        implemented for multigraphs; it exists only for consistency
+        with the other minimum spanning tree functions.
+
+    data : bool (default: True)
+        Flag for whether to yield edge attribute dicts.
+        If True, yield edges `(u, v, d)`, where `d` is the attribute dict.
+        If False, yield edges `(u, v)`.
+
+    ignore_nan : bool (default: False)
+        If a NaN is found as an edge weight normally an exception is raised.
+        If `ignore_nan is True` then that edge is ignored instead.
+
+    """
+    # Initialize a forest, assuming initially that it is the discrete
+    # partition of the nodes of the graph.
+    forest = UnionFind(G)
+
+    def best_edge(component):
+        """Returns the optimum (minimum or maximum) edge on the edge
+        boundary of the given set of nodes.
+
+        A return value of ``None`` indicates an empty boundary.
+
+        """
+        sign = 1 if minimum else -1
+        minwt = float("inf")
+        boundary = None
+        for e in nx.edge_boundary(G, component, data=True):
+            wt = e[-1].get(weight, 1) * sign
+            if isnan(wt):
+                if ignore_nan:
+                    continue
+                msg = f"NaN found as an edge weight. Edge {e}"
+                raise ValueError(msg)
+            if wt < minwt:
+                minwt = wt
+                boundary = e
+        return boundary
+
+    # Determine the optimum edge in the edge boundary of each component
+    # in the forest.
+    best_edges = (best_edge(component) for component in forest.to_sets())
+    best_edges = [edge for edge in best_edges if edge is not None]
+    # If each entry was ``None``, that means the graph was disconnected,
+    # so we are done generating the forest.
+    while best_edges:
+        # Determine the optimum edge in the edge boundary of each
+        # component in the forest.
+        #
+        # This must be a sequence, not an iterator. In this list, the
+        # same edge may appear twice, in different orientations (but
+        # that's okay, since a union operation will be called on the
+        # endpoints the first time it is seen, but not the second time).
+        #
+        # Any ``None`` indicates that the edge boundary for that
+        # component was empty, so that part of the forest has been
+        # completed.
+        #
+        # TODO This can be parallelized, both in the outer loop over
+        # each component in the forest and in the computation of the
+        # minimum. (Same goes for the identical lines outside the loop.)
+        best_edges = (best_edge(component) for component in forest.to_sets())
+        best_edges = [edge for edge in best_edges if edge is not None]
+        # Join trees in the forest using the best edges, and yield that
+        # edge, since it is part of the spanning tree.
+        #
+        # TODO This loop can be parallelized, to an extent (the union
+        # operation must be atomic).
+        for u, v, d in best_edges:
+            if forest[u] != forest[v]:
+                if data:
+                    yield u, v, d
+                else:
+                    yield u, v
+                forest.union(u, v)
+
+
+@nx._dispatchable(
+    edge_attrs={"weight": None, "partition": None}, preserve_edge_attrs="data"
+)
+def kruskal_mst_edges(
+    G, minimum, weight="weight", keys=True, data=True, ignore_nan=False, partition=None
+):
+    """
+    Iterate over edge of a Kruskal's algorithm min/max spanning tree.
+
+    Parameters
+    ----------
+    G : NetworkX Graph
+        The graph holding the tree of interest.
+
+    minimum : bool (default: True)
+        Find the minimum (True) or maximum (False) spanning tree.
+
+    weight : string (default: 'weight')
+        The name of the edge attribute holding the edge weights.
+
+    keys : bool (default: True)
+        If `G` is a multigraph, `keys` controls whether edge keys ar yielded.
+        Otherwise `keys` is ignored.
+
+    data : bool (default: True)
+        Flag for whether to yield edge attribute dicts.
+        If True, yield edges `(u, v, d)`, where `d` is the attribute dict.
+        If False, yield edges `(u, v)`.
+
+    ignore_nan : bool (default: False)
+        If a NaN is found as an edge weight normally an exception is raised.
+        If `ignore_nan is True` then that edge is ignored instead.
+
+    partition : string (default: None)
+        The name of the edge attribute holding the partition data, if it exists.
+        Partition data is written to the edges using the `EdgePartition` enum.
+        If a partition exists, all included edges and none of the excluded edges
+        will appear in the final tree. Open edges may or may not be used.
+
+    Yields
+    ------
+    edge tuple
+        The edges as discovered by Kruskal's method. Each edge can
+        take the following forms: `(u, v)`, `(u, v, d)` or `(u, v, k, d)`
+        depending on the `key` and `data` parameters
+    """
+    subtrees = UnionFind()
+    if G.is_multigraph():
+        edges = G.edges(keys=True, data=True)
+    else:
+        edges = G.edges(data=True)
+
+    """
+    Sort the edges of the graph with respect to the partition data. 
+    Edges are returned in the following order:
+
+    * Included edges
+    * Open edges from smallest to largest weight
+    * Excluded edges
+    """
+    included_edges = []
+    open_edges = []
+    for e in edges:
+        d = e[-1]
+        wt = d.get(weight, 1)
+        if isnan(wt):
+            if ignore_nan:
+                continue
+            raise ValueError(f"NaN found as an edge weight. Edge {e}")
+
+        edge = (wt,) + e
+        if d.get(partition) == EdgePartition.INCLUDED:
+            included_edges.append(edge)
+        elif d.get(partition) == EdgePartition.EXCLUDED:
+            continue
+        else:
+            open_edges.append(edge)
+
+    if minimum:
+        sorted_open_edges = sorted(open_edges, key=itemgetter(0))
+    else:
+        sorted_open_edges = sorted(open_edges, key=itemgetter(0), reverse=True)
+
+    # Condense the lists into one
+    included_edges.extend(sorted_open_edges)
+    sorted_edges = included_edges
+    del open_edges, sorted_open_edges, included_edges
+
+    # Multigraphs need to handle edge keys in addition to edge data.
+    if G.is_multigraph():
+        for wt, u, v, k, d in sorted_edges:
+            if subtrees[u] != subtrees[v]:
+                if keys:
+                    if data:
+                        yield u, v, k, d
+                    else:
+                        yield u, v, k
+                else:
+                    if data:
+                        yield u, v, d
+                    else:
+                        yield u, v
+                subtrees.union(u, v)
+    else:
+        for wt, u, v, d in sorted_edges:
+            if subtrees[u] != subtrees[v]:
+                if data:
+                    yield u, v, d
+                else:
+                    yield u, v
+                subtrees.union(u, v)
+
+
+@nx._dispatchable(edge_attrs="weight", preserve_edge_attrs="data")
+def prim_mst_edges(G, minimum, weight="weight", keys=True, data=True, ignore_nan=False):
+    """Iterate over edges of Prim's algorithm min/max spanning tree.
+
+    Parameters
+    ----------
+    G : NetworkX Graph
+        The graph holding the tree of interest.
+
+    minimum : bool (default: True)
+        Find the minimum (True) or maximum (False) spanning tree.
+
+    weight : string (default: 'weight')
+        The name of the edge attribute holding the edge weights.
+
+    keys : bool (default: True)
+        If `G` is a multigraph, `keys` controls whether edge keys ar yielded.
+        Otherwise `keys` is ignored.
+
+    data : bool (default: True)
+        Flag for whether to yield edge attribute dicts.
+        If True, yield edges `(u, v, d)`, where `d` is the attribute dict.
+        If False, yield edges `(u, v)`.
+
+    ignore_nan : bool (default: False)
+        If a NaN is found as an edge weight normally an exception is raised.
+        If `ignore_nan is True` then that edge is ignored instead.
+
+    """
+    is_multigraph = G.is_multigraph()
+    push = heappush
+    pop = heappop
+
+    nodes = set(G)
+    c = count()
+
+    sign = 1 if minimum else -1
+
+    while nodes:
+        u = nodes.pop()
+        frontier = []
+        visited = {u}
+        if is_multigraph:
+            for v, keydict in G.adj[u].items():
+                for k, d in keydict.items():
+                    wt = d.get(weight, 1) * sign
+                    if isnan(wt):
+                        if ignore_nan:
+                            continue
+                        msg = f"NaN found as an edge weight. Edge {(u, v, k, d)}"
+                        raise ValueError(msg)
+                    push(frontier, (wt, next(c), u, v, k, d))
+        else:
+            for v, d in G.adj[u].items():
+                wt = d.get(weight, 1) * sign
+                if isnan(wt):
+                    if ignore_nan:
+                        continue
+                    msg = f"NaN found as an edge weight. Edge {(u, v, d)}"
+                    raise ValueError(msg)
+                push(frontier, (wt, next(c), u, v, d))
+        while nodes and frontier:
+            if is_multigraph:
+                W, _, u, v, k, d = pop(frontier)
+            else:
+                W, _, u, v, d = pop(frontier)
+            if v in visited or v not in nodes:
+                continue
+            # Multigraphs need to handle edge keys in addition to edge data.
+            if is_multigraph and keys:
+                if data:
+                    yield u, v, k, d
+                else:
+                    yield u, v, k
+            else:
+                if data:
+                    yield u, v, d
+                else:
+                    yield u, v
+            # update frontier
+            visited.add(v)
+            nodes.discard(v)
+            if is_multigraph:
+                for w, keydict in G.adj[v].items():
+                    if w in visited:
+                        continue
+                    for k2, d2 in keydict.items():
+                        new_weight = d2.get(weight, 1) * sign
+                        if isnan(new_weight):
+                            if ignore_nan:
+                                continue
+                            msg = f"NaN found as an edge weight. Edge {(v, w, k2, d2)}"
+                            raise ValueError(msg)
+                        push(frontier, (new_weight, next(c), v, w, k2, d2))
+            else:
+                for w, d2 in G.adj[v].items():
+                    if w in visited:
+                        continue
+                    new_weight = d2.get(weight, 1) * sign
+                    if isnan(new_weight):
+                        if ignore_nan:
+                            continue
+                        msg = f"NaN found as an edge weight. Edge {(v, w, d2)}"
+                        raise ValueError(msg)
+                    push(frontier, (new_weight, next(c), v, w, d2))
+
+
+ALGORITHMS = {
+    "boruvka": boruvka_mst_edges,
+    "borůvka": boruvka_mst_edges,
+    "kruskal": kruskal_mst_edges,
+    "prim": prim_mst_edges,
+}
+
+
+@not_implemented_for("directed")
+@nx._dispatchable(edge_attrs="weight", preserve_edge_attrs="data")
+def minimum_spanning_edges(
+    G, algorithm="kruskal", weight="weight", keys=True, data=True, ignore_nan=False
+):
+    """Generate edges in a minimum spanning forest of an undirected
+    weighted graph.
+
+    A minimum spanning tree is a subgraph of the graph (a tree)
+    with the minimum sum of edge weights.  A spanning forest is a
+    union of the spanning trees for each connected component of the graph.
+
+    Parameters
+    ----------
+    G : undirected Graph
+       An undirected graph. If `G` is connected, then the algorithm finds a
+       spanning tree. Otherwise, a spanning forest is found.
+
+    algorithm : string
+       The algorithm to use when finding a minimum spanning tree. Valid
+       choices are 'kruskal', 'prim', or 'boruvka'. The default is 'kruskal'.
+
+    weight : string
+       Edge data key to use for weight (default 'weight').
+
+    keys : bool
+       Whether to yield edge key in multigraphs in addition to the edge.
+       If `G` is not a multigraph, this is ignored.
+
+    data : bool, optional
+       If True yield the edge data along with the edge.
+
+    ignore_nan : bool (default: False)
+        If a NaN is found as an edge weight normally an exception is raised.
+        If `ignore_nan is True` then that edge is ignored instead.
+
+    Returns
+    -------
+    edges : iterator
+       An iterator over edges in a maximum spanning tree of `G`.
+       Edges connecting nodes `u` and `v` are represented as tuples:
+       `(u, v, k, d)` or `(u, v, k)` or `(u, v, d)` or `(u, v)`
+
+       If `G` is a multigraph, `keys` indicates whether the edge key `k` will
+       be reported in the third position in the edge tuple. `data` indicates
+       whether the edge datadict `d` will appear at the end of the edge tuple.
+
+       If `G` is not a multigraph, the tuples are `(u, v, d)` if `data` is True
+       or `(u, v)` if `data` is False.
+
+    Examples
+    --------
+    >>> from networkx.algorithms import tree
+
+    Find minimum spanning edges by Kruskal's algorithm
+
+    >>> G = nx.cycle_graph(4)
+    >>> G.add_edge(0, 3, weight=2)
+    >>> mst = tree.minimum_spanning_edges(G, algorithm="kruskal", data=False)
+    >>> edgelist = list(mst)
+    >>> sorted(sorted(e) for e in edgelist)
+    [[0, 1], [1, 2], [2, 3]]
+
+    Find minimum spanning edges by Prim's algorithm
+
+    >>> G = nx.cycle_graph(4)
+    >>> G.add_edge(0, 3, weight=2)
+    >>> mst = tree.minimum_spanning_edges(G, algorithm="prim", data=False)
+    >>> edgelist = list(mst)
+    >>> sorted(sorted(e) for e in edgelist)
+    [[0, 1], [1, 2], [2, 3]]
+
+    Notes
+    -----
+    For Borůvka's algorithm, each edge must have a weight attribute, and
+    each edge weight must be distinct.
+
+    For the other algorithms, if the graph edges do not have a weight
+    attribute a default weight of 1 will be used.
+
+    Modified code from David Eppstein, April 2006
+    http://www.ics.uci.edu/~eppstein/PADS/
+
+    """
+    try:
+        algo = ALGORITHMS[algorithm]
+    except KeyError as err:
+        msg = f"{algorithm} is not a valid choice for an algorithm."
+        raise ValueError(msg) from err
+
+    return algo(
+        G, minimum=True, weight=weight, keys=keys, data=data, ignore_nan=ignore_nan
+    )
+
+
+@not_implemented_for("directed")
+@nx._dispatchable(edge_attrs="weight", preserve_edge_attrs="data")
+def maximum_spanning_edges(
+    G, algorithm="kruskal", weight="weight", keys=True, data=True, ignore_nan=False
+):
+    """Generate edges in a maximum spanning forest of an undirected
+    weighted graph.
+
+    A maximum spanning tree is a subgraph of the graph (a tree)
+    with the maximum possible sum of edge weights.  A spanning forest is a
+    union of the spanning trees for each connected component of the graph.
+
+    Parameters
+    ----------
+    G : undirected Graph
+       An undirected graph. If `G` is connected, then the algorithm finds a
+       spanning tree. Otherwise, a spanning forest is found.
+
+    algorithm : string
+       The algorithm to use when finding a maximum spanning tree. Valid
+       choices are 'kruskal', 'prim', or 'boruvka'. The default is 'kruskal'.
+
+    weight : string
+       Edge data key to use for weight (default 'weight').
+
+    keys : bool
+       Whether to yield edge key in multigraphs in addition to the edge.
+       If `G` is not a multigraph, this is ignored.
+
+    data : bool, optional
+       If True yield the edge data along with the edge.
+
+    ignore_nan : bool (default: False)
+        If a NaN is found as an edge weight normally an exception is raised.
+        If `ignore_nan is True` then that edge is ignored instead.
+
+    Returns
+    -------
+    edges : iterator
+       An iterator over edges in a maximum spanning tree of `G`.
+       Edges connecting nodes `u` and `v` are represented as tuples:
+       `(u, v, k, d)` or `(u, v, k)` or `(u, v, d)` or `(u, v)`
+
+       If `G` is a multigraph, `keys` indicates whether the edge key `k` will
+       be reported in the third position in the edge tuple. `data` indicates
+       whether the edge datadict `d` will appear at the end of the edge tuple.
+
+       If `G` is not a multigraph, the tuples are `(u, v, d)` if `data` is True
+       or `(u, v)` if `data` is False.
+
+    Examples
+    --------
+    >>> from networkx.algorithms import tree
+
+    Find maximum spanning edges by Kruskal's algorithm
+
+    >>> G = nx.cycle_graph(4)
+    >>> G.add_edge(0, 3, weight=2)
+    >>> mst = tree.maximum_spanning_edges(G, algorithm="kruskal", data=False)
+    >>> edgelist = list(mst)
+    >>> sorted(sorted(e) for e in edgelist)
+    [[0, 1], [0, 3], [1, 2]]
+
+    Find maximum spanning edges by Prim's algorithm
+
+    >>> G = nx.cycle_graph(4)
+    >>> G.add_edge(0, 3, weight=2)  # assign weight 2 to edge 0-3
+    >>> mst = tree.maximum_spanning_edges(G, algorithm="prim", data=False)
+    >>> edgelist = list(mst)
+    >>> sorted(sorted(e) for e in edgelist)
+    [[0, 1], [0, 3], [2, 3]]
+
+    Notes
+    -----
+    For Borůvka's algorithm, each edge must have a weight attribute, and
+    each edge weight must be distinct.
+
+    For the other algorithms, if the graph edges do not have a weight
+    attribute a default weight of 1 will be used.
+
+    Modified code from David Eppstein, April 2006
+    http://www.ics.uci.edu/~eppstein/PADS/
+    """
+    try:
+        algo = ALGORITHMS[algorithm]
+    except KeyError as err:
+        msg = f"{algorithm} is not a valid choice for an algorithm."
+        raise ValueError(msg) from err
+
+    return algo(
+        G, minimum=False, weight=weight, keys=keys, data=data, ignore_nan=ignore_nan
+    )
+
+
+@nx._dispatchable(preserve_all_attrs=True, returns_graph=True)
+def minimum_spanning_tree(G, weight="weight", algorithm="kruskal", ignore_nan=False):
+    """Returns a minimum spanning tree or forest on an undirected graph `G`.
+
+    Parameters
+    ----------
+    G : undirected graph
+        An undirected graph. If `G` is connected, then the algorithm finds a
+        spanning tree. Otherwise, a spanning forest is found.
+
+    weight : str
+       Data key to use for edge weights.
+
+    algorithm : string
+       The algorithm to use when finding a minimum spanning tree. Valid
+       choices are 'kruskal', 'prim', or 'boruvka'. The default is
+       'kruskal'.
+
+    ignore_nan : bool (default: False)
+        If a NaN is found as an edge weight normally an exception is raised.
+        If `ignore_nan is True` then that edge is ignored instead.
+
+    Returns
+    -------
+    G : NetworkX Graph
+       A minimum spanning tree or forest.
+
+    Examples
+    --------
+    >>> G = nx.cycle_graph(4)
+    >>> G.add_edge(0, 3, weight=2)
+    >>> T = nx.minimum_spanning_tree(G)
+    >>> sorted(T.edges(data=True))
+    [(0, 1, {}), (1, 2, {}), (2, 3, {})]
+
+
+    Notes
+    -----
+    For Borůvka's algorithm, each edge must have a weight attribute, and
+    each edge weight must be distinct.
+
+    For the other algorithms, if the graph edges do not have a weight
+    attribute a default weight of 1 will be used.
+
+    There may be more than one tree with the same minimum or maximum weight.
+    See :mod:`networkx.tree.recognition` for more detailed definitions.
+
+    Isolated nodes with self-loops are in the tree as edgeless isolated nodes.
+
+    """
+    edges = minimum_spanning_edges(
+        G, algorithm, weight, keys=True, data=True, ignore_nan=ignore_nan
+    )
+    T = G.__class__()  # Same graph class as G
+    T.graph.update(G.graph)
+    T.add_nodes_from(G.nodes.items())
+    T.add_edges_from(edges)
+    return T
+
+
+@nx._dispatchable(preserve_all_attrs=True, returns_graph=True)
+def partition_spanning_tree(
+    G, minimum=True, weight="weight", partition="partition", ignore_nan=False
+):
+    """
+    Find a spanning tree while respecting a partition of edges.
+
+    Edges can be flagged as either `INCLUDED` which are required to be in the
+    returned tree, `EXCLUDED`, which cannot be in the returned tree and `OPEN`.
+
+    This is used in the SpanningTreeIterator to create new partitions following
+    the algorithm of Sörensen and Janssens [1]_.
+
+    Parameters
+    ----------
+    G : undirected graph
+        An undirected graph.
+
+    minimum : bool (default: True)
+        Determines whether the returned tree is the minimum spanning tree of
+        the partition of the maximum one.
+
+    weight : str
+        Data key to use for edge weights.
+
+    partition : str
+        The key for the edge attribute containing the partition
+        data on the graph. Edges can be included, excluded or open using the
+        `EdgePartition` enum.
+
+    ignore_nan : bool (default: False)
+        If a NaN is found as an edge weight normally an exception is raised.
+        If `ignore_nan is True` then that edge is ignored instead.
+
+
+    Returns
+    -------
+    G : NetworkX Graph
+        A minimum spanning tree using all of the included edges in the graph and
+        none of the excluded edges.
+
+    References
+    ----------
+    .. [1] G.K. Janssens, K. Sörensen, An algorithm to generate all spanning
+           trees in order of increasing cost, Pesquisa Operacional, 2005-08,
+           Vol. 25 (2), p. 219-229,
+           https://www.scielo.br/j/pope/a/XHswBwRwJyrfL88dmMwYNWp/?lang=en
+    """
+    edges = kruskal_mst_edges(
+        G,
+        minimum,
+        weight,
+        keys=True,
+        data=True,
+        ignore_nan=ignore_nan,
+        partition=partition,
+    )
+    T = G.__class__()  # Same graph class as G
+    T.graph.update(G.graph)
+    T.add_nodes_from(G.nodes.items())
+    T.add_edges_from(edges)
+    return T
+
+
+@nx._dispatchable(preserve_all_attrs=True, returns_graph=True)
+def maximum_spanning_tree(G, weight="weight", algorithm="kruskal", ignore_nan=False):
+    """Returns a maximum spanning tree or forest on an undirected graph `G`.
+
+    Parameters
+    ----------
+    G : undirected graph
+        An undirected graph. If `G` is connected, then the algorithm finds a
+        spanning tree. Otherwise, a spanning forest is found.
+
+    weight : str
+       Data key to use for edge weights.
+
+    algorithm : string
+       The algorithm to use when finding a maximum spanning tree. Valid
+       choices are 'kruskal', 'prim', or 'boruvka'. The default is
+       'kruskal'.
+
+    ignore_nan : bool (default: False)
+        If a NaN is found as an edge weight normally an exception is raised.
+        If `ignore_nan is True` then that edge is ignored instead.
+
+
+    Returns
+    -------
+    G : NetworkX Graph
+       A maximum spanning tree or forest.
+
+
+    Examples
+    --------
+    >>> G = nx.cycle_graph(4)
+    >>> G.add_edge(0, 3, weight=2)
+    >>> T = nx.maximum_spanning_tree(G)
+    >>> sorted(T.edges(data=True))
+    [(0, 1, {}), (0, 3, {'weight': 2}), (1, 2, {})]
+
+
+    Notes
+    -----
+    For Borůvka's algorithm, each edge must have a weight attribute, and
+    each edge weight must be distinct.
+
+    For the other algorithms, if the graph edges do not have a weight
+    attribute a default weight of 1 will be used.
+
+    There may be more than one tree with the same minimum or maximum weight.
+    See :mod:`networkx.tree.recognition` for more detailed definitions.
+
+    Isolated nodes with self-loops are in the tree as edgeless isolated nodes.
+
+    """
+    edges = maximum_spanning_edges(
+        G, algorithm, weight, keys=True, data=True, ignore_nan=ignore_nan
+    )
+    edges = list(edges)
+    T = G.__class__()  # Same graph class as G
+    T.graph.update(G.graph)
+    T.add_nodes_from(G.nodes.items())
+    T.add_edges_from(edges)
+    return T
+
+
+@py_random_state(3)
+@nx._dispatchable(preserve_edge_attrs=True, returns_graph=True)
+def random_spanning_tree(G, weight=None, *, multiplicative=True, seed=None):
+    """
+    Sample a random spanning tree using the edges weights of `G`.
+
+    This function supports two different methods for determining the
+    probability of the graph. If ``multiplicative=True``, the probability
+    is based on the product of edge weights, and if ``multiplicative=False``
+    it is based on the sum of the edge weight. However, since it is
+    easier to determine the total weight of all spanning trees for the
+    multiplicative version, that is significantly faster and should be used if
+    possible. Additionally, setting `weight` to `None` will cause a spanning tree
+    to be selected with uniform probability.
+
+    The function uses algorithm A8 in [1]_ .
+
+    Parameters
+    ----------
+    G : nx.Graph
+        An undirected version of the original graph.
+
+    weight : string
+        The edge key for the edge attribute holding edge weight.
+
+    multiplicative : bool, default=True
+        If `True`, the probability of each tree is the product of its edge weight
+        over the sum of the product of all the spanning trees in the graph. If
+        `False`, the probability is the sum of its edge weight over the sum of
+        the sum of weights for all spanning trees in the graph.
+
+    seed : integer, random_state, or None (default)
+        Indicator of random number generation state.
+        See :ref:`Randomness<randomness>`.
+
+    Returns
+    -------
+    nx.Graph
+        A spanning tree using the distribution defined by the weight of the tree.
+
+    References
+    ----------
+    .. [1] V. Kulkarni, Generating random combinatorial objects, Journal of
+       Algorithms, 11 (1990), pp. 185–207
+    """
+
+    def find_node(merged_nodes, node):
+        """
+        We can think of clusters of contracted nodes as having one
+        representative in the graph. Each node which is not in merged_nodes
+        is still its own representative. Since a representative can be later
+        contracted, we need to recursively search though the dict to find
+        the final representative, but once we know it we can use path
+        compression to speed up the access of the representative for next time.
+
+        This cannot be replaced by the standard NetworkX union_find since that
+        data structure will merge nodes with less representing nodes into the
+        one with more representing nodes but this function requires we merge
+        them using the order that contract_edges contracts using.
+
+        Parameters
+        ----------
+        merged_nodes : dict
+            The dict storing the mapping from node to representative
+        node
+            The node whose representative we seek
+
+        Returns
+        -------
+        The representative of the `node`
+        """
+        if node not in merged_nodes:
+            return node
+        else:
+            rep = find_node(merged_nodes, merged_nodes[node])
+            merged_nodes[node] = rep
+            return rep
+
+    def prepare_graph():
+        """
+        For the graph `G`, remove all edges not in the set `V` and then
+        contract all edges in the set `U`.
+
+        Returns
+        -------
+        A copy of `G` which has had all edges not in `V` removed and all edges
+        in `U` contracted.
+        """
+
+        # The result is a MultiGraph version of G so that parallel edges are
+        # allowed during edge contraction
+        result = nx.MultiGraph(incoming_graph_data=G)
+
+        # Remove all edges not in V
+        edges_to_remove = set(result.edges()).difference(V)
+        result.remove_edges_from(edges_to_remove)
+
+        # Contract all edges in U
+        #
+        # Imagine that you have two edges to contract and they share an
+        # endpoint like this:
+        #                        [0] ----- [1] ----- [2]
+        # If we contract (0, 1) first, the contraction function will always
+        # delete the second node it is passed so the resulting graph would be
+        #                             [0] ----- [2]
+        # and edge (1, 2) no longer exists but (0, 2) would need to be contracted
+        # in its place now. That is why I use the below dict as a merge-find
+        # data structure with path compression to track how the nodes are merged.
+        merged_nodes = {}
+
+        for u, v in U:
+            u_rep = find_node(merged_nodes, u)
+            v_rep = find_node(merged_nodes, v)
+            # We cannot contract a node with itself
+            if u_rep == v_rep:
+                continue
+            nx.contracted_nodes(result, u_rep, v_rep, self_loops=False, copy=False)
+            merged_nodes[v_rep] = u_rep
+
+        return merged_nodes, result
+
+    def spanning_tree_total_weight(G, weight):
+        """
+        Find the sum of weights of the spanning trees of `G` using the
+        appropriate `method`.
+
+        This is easy if the chosen method is 'multiplicative', since we can
+        use Kirchhoff's Tree Matrix Theorem directly. However, with the
+        'additive' method, this process is slightly more complex and less
+        computationally efficient as we have to find the number of spanning
+        trees which contain each possible edge in the graph.
+
+        Parameters
+        ----------
+        G : NetworkX Graph
+            The graph to find the total weight of all spanning trees on.
+
+        weight : string
+            The key for the weight edge attribute of the graph.
+
+        Returns
+        -------
+        float
+            The sum of either the multiplicative or additive weight for all
+            spanning trees in the graph.
+        """
+        if multiplicative:
+            return nx.total_spanning_tree_weight(G, weight)
+        else:
+            # There are two cases for the total spanning tree additive weight.
+            # 1. There is one edge in the graph. Then the only spanning tree is
+            #    that edge itself, which will have a total weight of that edge
+            #    itself.
+            if G.number_of_edges() == 1:
+                return G.edges(data=weight).__iter__().__next__()[2]
+            # 2. There are no edges or two or more edges in the graph. Then, we find the
+            #    total weight of the spanning trees using the formula in the
+            #    reference paper: take the weight of each edge and multiply it by
+            #    the number of spanning trees which include that edge. This
+            #    can be accomplished by contracting the edge and finding the
+            #    multiplicative total spanning tree weight if the weight of each edge
+            #    is assumed to be 1, which is conveniently built into networkx already,
+            #    by calling total_spanning_tree_weight with weight=None.
+            #    Note that with no edges the returned value is just zero.
+            else:
+                total = 0
+                for u, v, w in G.edges(data=weight):
+                    total += w * nx.total_spanning_tree_weight(
+                        nx.contracted_edge(G, edge=(u, v), self_loops=False), None
+                    )
+                return total
+
+    if G.number_of_nodes() < 2:
+        # no edges in the spanning tree
+        return nx.empty_graph(G.nodes)
+
+    U = set()
+    st_cached_value = 0
+    V = set(G.edges())
+    shuffled_edges = list(G.edges())
+    seed.shuffle(shuffled_edges)
+
+    for u, v in shuffled_edges:
+        e_weight = G[u][v][weight] if weight is not None else 1
+        node_map, prepared_G = prepare_graph()
+        G_total_tree_weight = spanning_tree_total_weight(prepared_G, weight)
+        # Add the edge to U so that we can compute the total tree weight
+        # assuming we include that edge
+        # Now, if (u, v) cannot exist in G because it is fully contracted out
+        # of existence, then it by definition cannot influence G_e's Kirchhoff
+        # value. But, we also cannot pick it.
+        rep_edge = (find_node(node_map, u), find_node(node_map, v))
+        # Check to see if the 'representative edge' for the current edge is
+        # in prepared_G. If so, then we can pick it.
+        if rep_edge in prepared_G.edges:
+            prepared_G_e = nx.contracted_edge(
+                prepared_G, edge=rep_edge, self_loops=False
+            )
+            G_e_total_tree_weight = spanning_tree_total_weight(prepared_G_e, weight)
+            if multiplicative:
+                threshold = e_weight * G_e_total_tree_weight / G_total_tree_weight
+            else:
+                numerator = (
+                    st_cached_value + e_weight
+                ) * nx.total_spanning_tree_weight(prepared_G_e) + G_e_total_tree_weight
+                denominator = (
+                    st_cached_value * nx.total_spanning_tree_weight(prepared_G)
+                    + G_total_tree_weight
+                )
+                threshold = numerator / denominator
+        else:
+            threshold = 0.0
+        z = seed.uniform(0.0, 1.0)
+        if z > threshold:
+            # Remove the edge from V since we did not pick it.
+            V.remove((u, v))
+        else:
+            # Add the edge to U since we picked it.
+            st_cached_value += e_weight
+            U.add((u, v))
+        # If we decide to keep an edge, it may complete the spanning tree.
+        if len(U) == G.number_of_nodes() - 1:
+            spanning_tree = nx.Graph()
+            spanning_tree.add_edges_from(U)
+            return spanning_tree
+    raise Exception(f"Something went wrong! Only {len(U)} edges in the spanning tree!")
+
+
+class SpanningTreeIterator:
+    """
+    Iterate over all spanning trees of a graph in either increasing or
+    decreasing cost.
+
+    Notes
+    -----
+    This iterator uses the partition scheme from [1]_ (included edges,
+    excluded edges and open edges) as well as a modified Kruskal's Algorithm
+    to generate minimum spanning trees which respect the partition of edges.
+    For spanning trees with the same weight, ties are broken arbitrarily.
+
+    References
+    ----------
+    .. [1] G.K. Janssens, K. Sörensen, An algorithm to generate all spanning
+           trees in order of increasing cost, Pesquisa Operacional, 2005-08,
+           Vol. 25 (2), p. 219-229,
+           https://www.scielo.br/j/pope/a/XHswBwRwJyrfL88dmMwYNWp/?lang=en
+    """
+
+    @dataclass(order=True)
+    class Partition:
+        """
+        This dataclass represents a partition and stores a dict with the edge
+        data and the weight of the minimum spanning tree of the partition dict.
+        """
+
+        mst_weight: float
+        partition_dict: dict = field(compare=False)
+
+        def __copy__(self):
+            return SpanningTreeIterator.Partition(
+                self.mst_weight, self.partition_dict.copy()
+            )
+
+    def __init__(self, G, weight="weight", minimum=True, ignore_nan=False):
+        """
+        Initialize the iterator
+
+        Parameters
+        ----------
+        G : nx.Graph
+            The directed graph which we need to iterate trees over
+
+        weight : String, default = "weight"
+            The edge attribute used to store the weight of the edge
+
+        minimum : bool, default = True
+            Return the trees in increasing order while true and decreasing order
+            while false.
+
+        ignore_nan : bool, default = False
+            If a NaN is found as an edge weight normally an exception is raised.
+            If `ignore_nan is True` then that edge is ignored instead.
+        """
+        self.G = G.copy()
+        self.G.__networkx_cache__ = None  # Disable caching
+        self.weight = weight
+        self.minimum = minimum
+        self.ignore_nan = ignore_nan
+        # Randomly create a key for an edge attribute to hold the partition data
+        self.partition_key = (
+            "SpanningTreeIterators super secret partition attribute name"
+        )
+
+    def __iter__(self):
+        """
+        Returns
+        -------
+        SpanningTreeIterator
+            The iterator object for this graph
+        """
+        self.partition_queue = PriorityQueue()
+        self._clear_partition(self.G)
+        mst_weight = partition_spanning_tree(
+            self.G, self.minimum, self.weight, self.partition_key, self.ignore_nan
+        ).size(weight=self.weight)
+
+        self.partition_queue.put(
+            self.Partition(mst_weight if self.minimum else -mst_weight, {})
+        )
+
+        return self
+
+    def __next__(self):
+        """
+        Returns
+        -------
+        (multi)Graph
+            The spanning tree of next greatest weight, which ties broken
+            arbitrarily.
+        """
+        if self.partition_queue.empty():
+            del self.G, self.partition_queue
+            raise StopIteration
+
+        partition = self.partition_queue.get()
+        self._write_partition(partition)
+        next_tree = partition_spanning_tree(
+            self.G, self.minimum, self.weight, self.partition_key, self.ignore_nan
+        )
+        self._partition(partition, next_tree)
+
+        self._clear_partition(next_tree)
+        return next_tree
+
+    def _partition(self, partition, partition_tree):
+        """
+        Create new partitions based of the minimum spanning tree of the
+        current minimum partition.
+
+        Parameters
+        ----------
+        partition : Partition
+            The Partition instance used to generate the current minimum spanning
+            tree.
+        partition_tree : nx.Graph
+            The minimum spanning tree of the input partition.
+        """
+        # create two new partitions with the data from the input partition dict
+        p1 = self.Partition(0, partition.partition_dict.copy())
+        p2 = self.Partition(0, partition.partition_dict.copy())
+        for e in partition_tree.edges:
+            # determine if the edge was open or included
+            if e not in partition.partition_dict:
+                # This is an open edge
+                p1.partition_dict[e] = EdgePartition.EXCLUDED
+                p2.partition_dict[e] = EdgePartition.INCLUDED
+
+                self._write_partition(p1)
+                p1_mst = partition_spanning_tree(
+                    self.G,
+                    self.minimum,
+                    self.weight,
+                    self.partition_key,
+                    self.ignore_nan,
+                )
+                p1_mst_weight = p1_mst.size(weight=self.weight)
+                if nx.is_connected(p1_mst):
+                    p1.mst_weight = p1_mst_weight if self.minimum else -p1_mst_weight
+                    self.partition_queue.put(p1.__copy__())
+                p1.partition_dict = p2.partition_dict.copy()
+
+    def _write_partition(self, partition):
+        """
+        Writes the desired partition into the graph to calculate the minimum
+        spanning tree.
+
+        Parameters
+        ----------
+        partition : Partition
+            A Partition dataclass describing a partition on the edges of the
+            graph.
+        """
+
+        partition_dict = partition.partition_dict
+        partition_key = self.partition_key
+        G = self.G
+
+        edges = (
+            G.edges(keys=True, data=True) if G.is_multigraph() else G.edges(data=True)
+        )
+        for *e, d in edges:
+            d[partition_key] = partition_dict.get(tuple(e), EdgePartition.OPEN)
+
+    def _clear_partition(self, G):
+        """
+        Removes partition data from the graph
+        """
+        partition_key = self.partition_key
+        edges = (
+            G.edges(keys=True, data=True) if G.is_multigraph() else G.edges(data=True)
+        )
+        for *e, d in edges:
+            if partition_key in d:
+                del d[partition_key]
+
+
+@nx._dispatchable(edge_attrs="weight")
+def number_of_spanning_trees(G, *, root=None, weight=None):
+    """Returns the number of spanning trees in `G`.
+
+    A spanning tree for an undirected graph is a tree that connects
+    all nodes in the graph. For a directed graph, the analog of a
+    spanning tree is called a (spanning) arborescence. The arborescence
+    includes a unique directed path from the `root` node to each other node.
+    The graph must be weakly connected, and the root must be a node
+    that includes all nodes as successors [3]_. Note that to avoid
+    discussing sink-roots and reverse-arborescences, we have reversed
+    the edge orientation from [3]_ and use the in-degree laplacian.
+
+    This function (when `weight` is `None`) returns the number of
+    spanning trees for an undirected graph and the number of
+    arborescences from a single root node for a directed graph.
+    When `weight` is the name of an edge attribute which holds the
+    weight value of each edge, the function returns the sum over
+    all trees of the multiplicative weight of each tree. That is,
+    the weight of the tree is the product of its edge weights.
+
+    Kirchoff's Tree Matrix Theorem states that any cofactor of the
+    Laplacian matrix of a graph is the number of spanning trees in the
+    graph. (Here we use cofactors for a diagonal entry so that the
+    cofactor becomes the determinant of the matrix with one row
+    and its matching column removed.) For a weighted Laplacian matrix,
+    the cofactor is the sum across all spanning trees of the
+    multiplicative weight of each tree. That is, the weight of each
+    tree is the product of its edge weights. The theorem is also
+    known as Kirchhoff's theorem [1]_ and the Matrix-Tree theorem [2]_.
+
+    For directed graphs, a similar theorem (Tutte's Theorem) holds with
+    the cofactor chosen to be the one with row and column removed that
+    correspond to the root. The cofactor is the number of arborescences
+    with the specified node as root. And the weighted version gives the
+    sum of the arborescence weights with root `root`. The arborescence
+    weight is the product of its edge weights.
+
+    Parameters
+    ----------
+    G : NetworkX graph
+
+    root : node
+       A node in the directed graph `G` that has all nodes as descendants.
+       (This is ignored for undirected graphs.)
+
+    weight : string or None, optional (default=None)
+        The name of the edge attribute holding the edge weight.
+        If `None`, then each edge is assumed to have a weight of 1.
+
+    Returns
+    -------
+    Number
+        Undirected graphs:
+            The number of spanning trees of the graph `G`.
+            Or the sum of all spanning tree weights of the graph `G`
+            where the weight of a tree is the product of its edge weights.
+        Directed graphs:
+            The number of arborescences of `G` rooted at node `root`.
+            Or the sum of all arborescence weights of the graph `G` with
+            specified root where the weight of an arborescence is the product
+            of its edge weights.
+
+    Raises
+    ------
+    NetworkXPointlessConcept
+        If `G` does not contain any nodes.
+
+    NetworkXError
+        If the graph `G` is directed and the root node
+        is not specified or is not in G.
+
+    Examples
+    --------
+    >>> G = nx.complete_graph(5)
+    >>> round(nx.number_of_spanning_trees(G))
+    125
+
+    >>> G = nx.Graph()
+    >>> G.add_edge(1, 2, weight=2)
+    >>> G.add_edge(1, 3, weight=1)
+    >>> G.add_edge(2, 3, weight=1)
+    >>> round(nx.number_of_spanning_trees(G, weight="weight"))
+    5
+
+    Notes
+    -----
+    Self-loops are excluded. Multi-edges are contracted in one edge
+    equal to the sum of the weights.
+
+    References
+    ----------
+    .. [1] Wikipedia
+       "Kirchhoff's theorem."
+       https://en.wikipedia.org/wiki/Kirchhoff%27s_theorem
+    .. [2] Kirchhoff, G. R.
+        Über die Auflösung der Gleichungen, auf welche man
+        bei der Untersuchung der linearen Vertheilung
+        Galvanischer Ströme geführt wird
+        Annalen der Physik und Chemie, vol. 72, pp. 497-508, 1847.
+    .. [3] Margoliash, J.
+        "Matrix-Tree Theorem for Directed Graphs"
+        https://www.math.uchicago.edu/~may/VIGRE/VIGRE2010/REUPapers/Margoliash.pdf
+    """
+    import numpy as np
+
+    if len(G) == 0:
+        raise nx.NetworkXPointlessConcept("Graph G must contain at least one node.")
+
+    # undirected G
+    if not nx.is_directed(G):
+        if not nx.is_connected(G):
+            return 0
+        G_laplacian = nx.laplacian_matrix(G, weight=weight).toarray()
+        return float(np.linalg.det(G_laplacian[1:, 1:]))
+
+    # directed G
+    if root is None:
+        raise nx.NetworkXError("Input `root` must be provided when G is directed")
+    if root not in G:
+        raise nx.NetworkXError("The node root is not in the graph G.")
+    if not nx.is_weakly_connected(G):
+        return 0
+
+    # Compute directed Laplacian matrix
+    nodelist = [root] + [n for n in G if n != root]
+    A = nx.adjacency_matrix(G, nodelist=nodelist, weight=weight)
+    D = np.diag(A.sum(axis=0))
+    G_laplacian = D - A
+
+    # Compute number of spanning trees
+    return float(np.linalg.det(G_laplacian[1:, 1:]))
diff --git a/.venv/lib/python3.12/site-packages/networkx/algorithms/tree/operations.py b/.venv/lib/python3.12/site-packages/networkx/algorithms/tree/operations.py
new file mode 100644
index 00000000..6c3e8394
--- /dev/null
+++ b/.venv/lib/python3.12/site-packages/networkx/algorithms/tree/operations.py
@@ -0,0 +1,105 @@
+"""Operations on trees."""
+
+from functools import partial
+from itertools import accumulate, chain
+
+import networkx as nx
+
+__all__ = ["join_trees"]
+
+
+# Argument types don't match dispatching, but allow manual selection of backend
+@nx._dispatchable(graphs=None, returns_graph=True)
+def join_trees(rooted_trees, *, label_attribute=None, first_label=0):
+    """Returns a new rooted tree made by joining `rooted_trees`
+
+    Constructs a new tree by joining each tree in `rooted_trees`.
+    A new root node is added and connected to each of the roots
+    of the input trees. While copying the nodes from the trees,
+    relabeling to integers occurs. If the `label_attribute` is provided,
+    the old node labels will be stored in the new tree under this attribute.
+
+    Parameters
+    ----------
+    rooted_trees : list
+        A list of pairs in which each left element is a NetworkX graph
+        object representing a tree and each right element is the root
+        node of that tree. The nodes of these trees will be relabeled to
+        integers.
+
+    label_attribute : str
+        If provided, the old node labels will be stored in the new tree
+        under this node attribute. If not provided, the original labels
+        of the nodes in the input trees are not stored.
+
+    first_label : int, optional (default=0)
+        Specifies the label for the new root node. If provided, the root node of the joined tree
+        will have this label. If not provided, the root node will default to a label of 0.
+
+    Returns
+    -------
+    NetworkX graph
+        The rooted tree resulting from joining the provided `rooted_trees`. The new tree has a root node
+        labeled as specified by `first_label` (defaulting to 0 if not provided). Subtrees from the input
+        `rooted_trees` are attached to this new root node. Each non-root node, if the `label_attribute`
+        is provided, has an attribute that indicates the original label of the node in the input tree.
+
+    Notes
+    -----
+    Trees are stored in NetworkX as NetworkX Graphs. There is no specific
+    enforcement of the fact that these are trees. Testing for each tree
+    can be done using :func:`networkx.is_tree`.
+
+    Graph, edge, and node attributes are propagated from the given
+    rooted trees to the created tree. If there are any overlapping graph
+    attributes, those from later trees will overwrite those from earlier
+    trees in the tuple of positional arguments.
+
+    Examples
+    --------
+    Join two full balanced binary trees of height *h* to get a full
+    balanced binary tree of depth *h* + 1::
+
+        >>> h = 4
+        >>> left = nx.balanced_tree(2, h)
+        >>> right = nx.balanced_tree(2, h)
+        >>> joined_tree = nx.join_trees([(left, 0), (right, 0)])
+        >>> nx.is_isomorphic(joined_tree, nx.balanced_tree(2, h + 1))
+        True
+
+    """
+    if not rooted_trees:
+        return nx.empty_graph(1)
+
+    # Unzip the zipped list of (tree, root) pairs.
+    trees, roots = zip(*rooted_trees)
+
+    # The join of the trees has the same type as the type of the first tree.
+    R = type(trees[0])()
+
+    lengths = (len(tree) for tree in trees[:-1])
+    first_labels = list(accumulate(lengths, initial=first_label + 1))
+
+    new_roots = []
+    for tree, root, first_node in zip(trees, roots, first_labels):
+        new_root = first_node + list(tree.nodes()).index(root)
+        new_roots.append(new_root)
+
+    # Relabel the nodes so that their union is the integers starting at first_label.
+    relabel = partial(
+        nx.convert_node_labels_to_integers, label_attribute=label_attribute
+    )
+    new_trees = [
+        relabel(tree, first_label=first_label)
+        for tree, first_label in zip(trees, first_labels)
+    ]
+
+    # Add all sets of nodes and edges, attributes
+    for tree in new_trees:
+        R.update(tree)
+
+    # Finally, join the subtrees at the root. We know first_label is unused by the way we relabeled the subtrees.
+    R.add_node(first_label)
+    R.add_edges_from((first_label, root) for root in new_roots)
+
+    return R
diff --git a/.venv/lib/python3.12/site-packages/networkx/algorithms/tree/recognition.py b/.venv/lib/python3.12/site-packages/networkx/algorithms/tree/recognition.py
new file mode 100644
index 00000000..a9eae987
--- /dev/null
+++ b/.venv/lib/python3.12/site-packages/networkx/algorithms/tree/recognition.py
@@ -0,0 +1,273 @@
+"""
+Recognition Tests
+=================
+
+A *forest* is an acyclic, undirected graph, and a *tree* is a connected forest.
+Depending on the subfield, there are various conventions for generalizing these
+definitions to directed graphs.
+
+In one convention, directed variants of forest and tree are defined in an
+identical manner, except that the direction of the edges is ignored. In effect,
+each directed edge is treated as a single undirected edge. Then, additional
+restrictions are imposed to define *branchings* and *arborescences*.
+
+In another convention, directed variants of forest and tree correspond to
+the previous convention's branchings and arborescences, respectively. Then two
+new terms, *polyforest* and *polytree*, are defined to correspond to the other
+convention's forest and tree.
+
+Summarizing::
+
+   +-----------------------------+
+   | Convention A | Convention B |
+   +=============================+
+   | forest       | polyforest   |
+   | tree         | polytree     |
+   | branching    | forest       |
+   | arborescence | tree         |
+   +-----------------------------+
+
+Each convention has its reasons. The first convention emphasizes definitional
+similarity in that directed forests and trees are only concerned with
+acyclicity and do not have an in-degree constraint, just as their undirected
+counterparts do not. The second convention emphasizes functional similarity
+in the sense that the directed analog of a spanning tree is a spanning
+arborescence. That is, take any spanning tree and choose one node as the root.
+Then every edge is assigned a direction such there is a directed path from the
+root to every other node. The result is a spanning arborescence.
+
+NetworkX follows convention "A". Explicitly, these are:
+
+undirected forest
+   An undirected graph with no undirected cycles.
+
+undirected tree
+   A connected, undirected forest.
+
+directed forest
+   A directed graph with no undirected cycles. Equivalently, the underlying
+   graph structure (which ignores edge orientations) is an undirected forest.
+   In convention B, this is known as a polyforest.
+
+directed tree
+   A weakly connected, directed forest. Equivalently, the underlying graph
+   structure (which ignores edge orientations) is an undirected tree. In
+   convention B, this is known as a polytree.
+
+branching
+   A directed forest with each node having, at most, one parent. So the maximum
+   in-degree is equal to 1. In convention B, this is known as a forest.
+
+arborescence
+   A directed tree with each node having, at most, one parent. So the maximum
+   in-degree is equal to 1. In convention B, this is known as a tree.
+
+For trees and arborescences, the adjective "spanning" may be added to designate
+that the graph, when considered as a forest/branching, consists of a single
+tree/arborescence that includes all nodes in the graph. It is true, by
+definition, that every tree/arborescence is spanning with respect to the nodes
+that define the tree/arborescence and so, it might seem redundant to introduce
+the notion of "spanning". However, the nodes may represent a subset of
+nodes from a larger graph, and it is in this context that the term "spanning"
+becomes a useful notion.
+
+"""
+
+import networkx as nx
+
+__all__ = ["is_arborescence", "is_branching", "is_forest", "is_tree"]
+
+
+@nx.utils.not_implemented_for("undirected")
+@nx._dispatchable
+def is_arborescence(G):
+    """
+    Returns True if `G` is an arborescence.
+
+    An arborescence is a directed tree with maximum in-degree equal to 1.
+
+    Parameters
+    ----------
+    G : graph
+        The graph to test.
+
+    Returns
+    -------
+    b : bool
+        A boolean that is True if `G` is an arborescence.
+
+    Examples
+    --------
+    >>> G = nx.DiGraph([(0, 1), (0, 2), (2, 3), (3, 4)])
+    >>> nx.is_arborescence(G)
+    True
+    >>> G.remove_edge(0, 1)
+    >>> G.add_edge(1, 2)  # maximum in-degree is 2
+    >>> nx.is_arborescence(G)
+    False
+
+    Notes
+    -----
+    In another convention, an arborescence is known as a *tree*.
+
+    See Also
+    --------
+    is_tree
+
+    """
+    return is_tree(G) and max(d for n, d in G.in_degree()) <= 1
+
+
+@nx.utils.not_implemented_for("undirected")
+@nx._dispatchable
+def is_branching(G):
+    """
+    Returns True if `G` is a branching.
+
+    A branching is a directed forest with maximum in-degree equal to 1.
+
+    Parameters
+    ----------
+    G : directed graph
+        The directed graph to test.
+
+    Returns
+    -------
+    b : bool
+        A boolean that is True if `G` is a branching.
+
+    Examples
+    --------
+    >>> G = nx.DiGraph([(0, 1), (1, 2), (2, 3), (3, 4)])
+    >>> nx.is_branching(G)
+    True
+    >>> G.remove_edge(2, 3)
+    >>> G.add_edge(3, 1)  # maximum in-degree is 2
+    >>> nx.is_branching(G)
+    False
+
+    Notes
+    -----
+    In another convention, a branching is also known as a *forest*.
+
+    See Also
+    --------
+    is_forest
+
+    """
+    return is_forest(G) and max(d for n, d in G.in_degree()) <= 1
+
+
+@nx._dispatchable
+def is_forest(G):
+    """
+    Returns True if `G` is a forest.
+
+    A forest is a graph with no undirected cycles.
+
+    For directed graphs, `G` is a forest if the underlying graph is a forest.
+    The underlying graph is obtained by treating each directed edge as a single
+    undirected edge in a multigraph.
+
+    Parameters
+    ----------
+    G : graph
+        The graph to test.
+
+    Returns
+    -------
+    b : bool
+        A boolean that is True if `G` is a forest.
+
+    Raises
+    ------
+    NetworkXPointlessConcept
+        If `G` is empty.
+
+    Examples
+    --------
+    >>> G = nx.Graph()
+    >>> G.add_edges_from([(1, 2), (1, 3), (2, 4), (2, 5)])
+    >>> nx.is_forest(G)
+    True
+    >>> G.add_edge(4, 1)
+    >>> nx.is_forest(G)
+    False
+
+    Notes
+    -----
+    In another convention, a directed forest is known as a *polyforest* and
+    then *forest* corresponds to a *branching*.
+
+    See Also
+    --------
+    is_branching
+
+    """
+    if len(G) == 0:
+        raise nx.exception.NetworkXPointlessConcept("G has no nodes.")
+
+    if G.is_directed():
+        components = (G.subgraph(c) for c in nx.weakly_connected_components(G))
+    else:
+        components = (G.subgraph(c) for c in nx.connected_components(G))
+
+    return all(len(c) - 1 == c.number_of_edges() for c in components)
+
+
+@nx._dispatchable
+def is_tree(G):
+    """
+    Returns True if `G` is a tree.
+
+    A tree is a connected graph with no undirected cycles.
+
+    For directed graphs, `G` is a tree if the underlying graph is a tree. The
+    underlying graph is obtained by treating each directed edge as a single
+    undirected edge in a multigraph.
+
+    Parameters
+    ----------
+    G : graph
+        The graph to test.
+
+    Returns
+    -------
+    b : bool
+        A boolean that is True if `G` is a tree.
+
+    Raises
+    ------
+    NetworkXPointlessConcept
+        If `G` is empty.
+
+    Examples
+    --------
+    >>> G = nx.Graph()
+    >>> G.add_edges_from([(1, 2), (1, 3), (2, 4), (2, 5)])
+    >>> nx.is_tree(G)  # n-1 edges
+    True
+    >>> G.add_edge(3, 4)
+    >>> nx.is_tree(G)  # n edges
+    False
+
+    Notes
+    -----
+    In another convention, a directed tree is known as a *polytree* and then
+    *tree* corresponds to an *arborescence*.
+
+    See Also
+    --------
+    is_arborescence
+
+    """
+    if len(G) == 0:
+        raise nx.exception.NetworkXPointlessConcept("G has no nodes.")
+
+    if G.is_directed():
+        is_connected = nx.is_weakly_connected
+    else:
+        is_connected = nx.is_connected
+
+    # A connected graph with no cycles has n-1 edges.
+    return len(G) - 1 == G.number_of_edges() and is_connected(G)
diff --git a/.venv/lib/python3.12/site-packages/networkx/algorithms/tree/tests/__init__.py b/.venv/lib/python3.12/site-packages/networkx/algorithms/tree/tests/__init__.py
new file mode 100644
index 00000000..e69de29b
--- /dev/null
+++ b/.venv/lib/python3.12/site-packages/networkx/algorithms/tree/tests/__init__.py
diff --git a/.venv/lib/python3.12/site-packages/networkx/algorithms/tree/tests/test_branchings.py b/.venv/lib/python3.12/site-packages/networkx/algorithms/tree/tests/test_branchings.py
new file mode 100644
index 00000000..e19ddee3
--- /dev/null
+++ b/.venv/lib/python3.12/site-packages/networkx/algorithms/tree/tests/test_branchings.py
@@ -0,0 +1,624 @@
+import math
+from operator import itemgetter
+
+import pytest
+
+np = pytest.importorskip("numpy")
+
+import networkx as nx
+from networkx.algorithms.tree import branchings, recognition
+
+#
+# Explicitly discussed examples from Edmonds paper.
+#
+
+# Used in Figures A-F.
+#
+# fmt: off
+G_array = np.array([
+    # 0   1   2   3   4   5   6   7   8
+    [0, 0, 12, 0, 12, 0, 0, 0, 0],  # 0
+    [4, 0, 0, 0, 0, 13, 0, 0, 0],  # 1
+    [0, 17, 0, 21, 0, 12, 0, 0, 0],  # 2
+    [5, 0, 0, 0, 17, 0, 18, 0, 0],  # 3
+    [0, 0, 0, 0, 0, 0, 0, 12, 0],  # 4
+    [0, 0, 0, 0, 0, 0, 14, 0, 12],  # 5
+    [0, 0, 21, 0, 0, 0, 0, 0, 15],  # 6
+    [0, 0, 0, 19, 0, 0, 15, 0, 0],  # 7
+    [0, 0, 0, 0, 0, 0, 0, 18, 0],  # 8
+], dtype=int)
+
+# Two copies of the graph from the original paper as disconnected components
+G_big_array = np.zeros(np.array(G_array.shape) * 2, dtype=int)
+G_big_array[:G_array.shape[0], :G_array.shape[1]] = G_array
+G_big_array[G_array.shape[0]:, G_array.shape[1]:] = G_array
+
+# fmt: on
+
+
+def G1():
+    G = nx.from_numpy_array(G_array, create_using=nx.MultiDiGraph)
+    return G
+
+
+def G2():
+    # Now we shift all the weights by -10.
+    # Should not affect optimal arborescence, but does affect optimal branching.
+    Garr = G_array.copy()
+    Garr[np.nonzero(Garr)] -= 10
+    G = nx.from_numpy_array(Garr, create_using=nx.MultiDiGraph)
+    return G
+
+
+# An optimal branching for G1 that is also a spanning arborescence. So it is
+# also an optimal spanning arborescence.
+#
+optimal_arborescence_1 = [
+    (0, 2, 12),
+    (2, 1, 17),
+    (2, 3, 21),
+    (1, 5, 13),
+    (3, 4, 17),
+    (3, 6, 18),
+    (6, 8, 15),
+    (8, 7, 18),
+]
+
+# For G2, the optimal branching of G1 (with shifted weights) is no longer
+# an optimal branching, but it is still an optimal spanning arborescence
+# (just with shifted weights). An optimal branching for G2 is similar to what
+# appears in figure G (this is greedy_subopt_branching_1a below), but with the
+# edge (3, 0, 5), which is now (3, 0, -5), removed. Thus, the optimal branching
+# is not a spanning arborescence. The code finds optimal_branching_2a.
+# An alternative and equivalent branching is optimal_branching_2b. We would
+# need to modify the code to iterate through all equivalent optimal branchings.
+#
+# These are maximal branchings or arborescences.
+optimal_branching_2a = [
+    (5, 6, 4),
+    (6, 2, 11),
+    (6, 8, 5),
+    (8, 7, 8),
+    (2, 1, 7),
+    (2, 3, 11),
+    (3, 4, 7),
+]
+optimal_branching_2b = [
+    (8, 7, 8),
+    (7, 3, 9),
+    (3, 4, 7),
+    (3, 6, 8),
+    (6, 2, 11),
+    (2, 1, 7),
+    (1, 5, 3),
+]
+optimal_arborescence_2 = [
+    (0, 2, 2),
+    (2, 1, 7),
+    (2, 3, 11),
+    (1, 5, 3),
+    (3, 4, 7),
+    (3, 6, 8),
+    (6, 8, 5),
+    (8, 7, 8),
+]
+
+# Two suboptimal maximal branchings on G1 obtained from a greedy algorithm.
+# 1a matches what is shown in Figure G in Edmonds's paper.
+greedy_subopt_branching_1a = [
+    (5, 6, 14),
+    (6, 2, 21),
+    (6, 8, 15),
+    (8, 7, 18),
+    (2, 1, 17),
+    (2, 3, 21),
+    (3, 0, 5),
+    (3, 4, 17),
+]
+greedy_subopt_branching_1b = [
+    (8, 7, 18),
+    (7, 6, 15),
+    (6, 2, 21),
+    (2, 1, 17),
+    (2, 3, 21),
+    (1, 5, 13),
+    (3, 0, 5),
+    (3, 4, 17),
+]
+
+
+def build_branching(edges, double=False):
+    G = nx.DiGraph()
+    for u, v, weight in edges:
+        G.add_edge(u, v, weight=weight)
+        if double:
+            G.add_edge(u + 9, v + 9, weight=weight)
+    return G
+
+
+def sorted_edges(G, attr="weight", default=1):
+    edges = [(u, v, data.get(attr, default)) for (u, v, data) in G.edges(data=True)]
+    edges = sorted(edges, key=lambda x: (x[2], x[1], x[0]))
+    return edges
+
+
+def assert_equal_branchings(G1, G2, attr="weight", default=1):
+    edges1 = list(G1.edges(data=True))
+    edges2 = list(G2.edges(data=True))
+    assert len(edges1) == len(edges2)
+
+    # Grab the weights only.
+    e1 = sorted_edges(G1, attr, default)
+    e2 = sorted_edges(G2, attr, default)
+
+    for a, b in zip(e1, e2):
+        assert a[:2] == b[:2]
+        np.testing.assert_almost_equal(a[2], b[2])
+
+
+################
+
+
+def test_optimal_branching1():
+    G = build_branching(optimal_arborescence_1)
+    assert recognition.is_arborescence(G), True
+    assert branchings.branching_weight(G) == 131
+
+
+def test_optimal_branching2a():
+    G = build_branching(optimal_branching_2a)
+    assert recognition.is_arborescence(G), True
+    assert branchings.branching_weight(G) == 53
+
+
+def test_optimal_branching2b():
+    G = build_branching(optimal_branching_2b)
+    assert recognition.is_arborescence(G), True
+    assert branchings.branching_weight(G) == 53
+
+
+def test_optimal_arborescence2():
+    G = build_branching(optimal_arborescence_2)
+    assert recognition.is_arborescence(G), True
+    assert branchings.branching_weight(G) == 51
+
+
+def test_greedy_suboptimal_branching1a():
+    G = build_branching(greedy_subopt_branching_1a)
+    assert recognition.is_arborescence(G), True
+    assert branchings.branching_weight(G) == 128
+
+
+def test_greedy_suboptimal_branching1b():
+    G = build_branching(greedy_subopt_branching_1b)
+    assert recognition.is_arborescence(G), True
+    assert branchings.branching_weight(G) == 127
+
+
+def test_greedy_max1():
+    # Standard test.
+    #
+    G = G1()
+    B = branchings.greedy_branching(G)
+    # There are only two possible greedy branchings. The sorting is such
+    # that it should equal the second suboptimal branching: 1b.
+    B_ = build_branching(greedy_subopt_branching_1b)
+    assert_equal_branchings(B, B_)
+
+
+def test_greedy_branching_kwarg_kind():
+    G = G1()
+    with pytest.raises(nx.NetworkXException, match="Unknown value for `kind`."):
+        B = branchings.greedy_branching(G, kind="lol")
+
+
+def test_greedy_branching_for_unsortable_nodes():
+    G = nx.DiGraph()
+    G.add_weighted_edges_from([((2, 3), 5, 1), (3, "a", 1), (2, 4, 5)])
+    edges = [(u, v, data.get("weight", 1)) for (u, v, data) in G.edges(data=True)]
+    with pytest.raises(TypeError):
+        edges.sort(key=itemgetter(2, 0, 1), reverse=True)
+    B = branchings.greedy_branching(G, kind="max").edges(data=True)
+    assert list(B) == [
+        ((2, 3), 5, {"weight": 1}),
+        (3, "a", {"weight": 1}),
+        (2, 4, {"weight": 5}),
+    ]
+
+
+def test_greedy_max2():
+    # Different default weight.
+    #
+    G = G1()
+    del G[1][0][0]["weight"]
+    B = branchings.greedy_branching(G, default=6)
+    # Chosen so that edge (3,0,5) is not selected and (1,0,6) is instead.
+
+    edges = [
+        (1, 0, 6),
+        (1, 5, 13),
+        (7, 6, 15),
+        (2, 1, 17),
+        (3, 4, 17),
+        (8, 7, 18),
+        (2, 3, 21),
+        (6, 2, 21),
+    ]
+    B_ = build_branching(edges)
+    assert_equal_branchings(B, B_)
+
+
+def test_greedy_max3():
+    # All equal weights.
+    #
+    G = G1()
+    B = branchings.greedy_branching(G, attr=None)
+
+    # This is mostly arbitrary...the output was generated by running the algo.
+    edges = [
+        (2, 1, 1),
+        (3, 0, 1),
+        (3, 4, 1),
+        (5, 8, 1),
+        (6, 2, 1),
+        (7, 3, 1),
+        (7, 6, 1),
+        (8, 7, 1),
+    ]
+    B_ = build_branching(edges)
+    assert_equal_branchings(B, B_, default=1)
+
+
+def test_greedy_min():
+    G = G1()
+    B = branchings.greedy_branching(G, kind="min")
+
+    edges = [
+        (1, 0, 4),
+        (0, 2, 12),
+        (0, 4, 12),
+        (2, 5, 12),
+        (4, 7, 12),
+        (5, 8, 12),
+        (5, 6, 14),
+        (7, 3, 19),
+    ]
+    B_ = build_branching(edges)
+    assert_equal_branchings(B, B_)
+
+
+def test_edmonds1_maxbranch():
+    G = G1()
+    x = branchings.maximum_branching(G)
+    x_ = build_branching(optimal_arborescence_1)
+    assert_equal_branchings(x, x_)
+
+
+def test_edmonds1_maxarbor():
+    G = G1()
+    x = branchings.maximum_spanning_arborescence(G)
+    x_ = build_branching(optimal_arborescence_1)
+    assert_equal_branchings(x, x_)
+
+
+def test_edmonds1_minimal_branching():
+    # graph will have something like a minimum arborescence but no spanning one
+    G = nx.from_numpy_array(G_big_array, create_using=nx.DiGraph)
+    B = branchings.minimal_branching(G)
+    edges = [
+        (3, 0, 5),
+        (0, 2, 12),
+        (0, 4, 12),
+        (2, 5, 12),
+        (4, 7, 12),
+        (5, 8, 12),
+        (5, 6, 14),
+        (2, 1, 17),
+    ]
+    B_ = build_branching(edges, double=True)
+    assert_equal_branchings(B, B_)
+
+
+def test_edmonds2_maxbranch():
+    G = G2()
+    x = branchings.maximum_branching(G)
+    x_ = build_branching(optimal_branching_2a)
+    assert_equal_branchings(x, x_)
+
+
+def test_edmonds2_maxarbor():
+    G = G2()
+    x = branchings.maximum_spanning_arborescence(G)
+    x_ = build_branching(optimal_arborescence_2)
+    assert_equal_branchings(x, x_)
+
+
+def test_edmonds2_minarbor():
+    G = G1()
+    x = branchings.minimum_spanning_arborescence(G)
+    # This was obtained from algorithm. Need to verify it independently.
+    # Branch weight is: 96
+    edges = [
+        (3, 0, 5),
+        (0, 2, 12),
+        (0, 4, 12),
+        (2, 5, 12),
+        (4, 7, 12),
+        (5, 8, 12),
+        (5, 6, 14),
+        (2, 1, 17),
+    ]
+    x_ = build_branching(edges)
+    assert_equal_branchings(x, x_)
+
+
+def test_edmonds3_minbranch1():
+    G = G1()
+    x = branchings.minimum_branching(G)
+    edges = []
+    x_ = build_branching(edges)
+    assert_equal_branchings(x, x_)
+
+
+def test_edmonds3_minbranch2():
+    G = G1()
+    G.add_edge(8, 9, weight=-10)
+    x = branchings.minimum_branching(G)
+    edges = [(8, 9, -10)]
+    x_ = build_branching(edges)
+    assert_equal_branchings(x, x_)
+
+
+# Need more tests
+
+
+def test_mst():
+    # Make sure we get the same results for undirected graphs.
+    # Example from: https://en.wikipedia.org/wiki/Kruskal's_algorithm
+    G = nx.Graph()
+    edgelist = [
+        (0, 3, [("weight", 5)]),
+        (0, 1, [("weight", 7)]),
+        (1, 3, [("weight", 9)]),
+        (1, 2, [("weight", 8)]),
+        (1, 4, [("weight", 7)]),
+        (3, 4, [("weight", 15)]),
+        (3, 5, [("weight", 6)]),
+        (2, 4, [("weight", 5)]),
+        (4, 5, [("weight", 8)]),
+        (4, 6, [("weight", 9)]),
+        (5, 6, [("weight", 11)]),
+    ]
+    G.add_edges_from(edgelist)
+    G = G.to_directed()
+    x = branchings.minimum_spanning_arborescence(G)
+
+    edges = [
+        ({0, 1}, 7),
+        ({0, 3}, 5),
+        ({3, 5}, 6),
+        ({1, 4}, 7),
+        ({4, 2}, 5),
+        ({4, 6}, 9),
+    ]
+
+    assert x.number_of_edges() == len(edges)
+    for u, v, d in x.edges(data=True):
+        assert ({u, v}, d["weight"]) in edges
+
+
+def test_mixed_nodetypes():
+    # Smoke test to make sure no TypeError is raised for mixed node types.
+    G = nx.Graph()
+    edgelist = [(0, 3, [("weight", 5)]), (0, "1", [("weight", 5)])]
+    G.add_edges_from(edgelist)
+    G = G.to_directed()
+    x = branchings.minimum_spanning_arborescence(G)
+
+
+def test_edmonds1_minbranch():
+    # Using -G_array and min should give the same as optimal_arborescence_1,
+    # but with all edges negative.
+    edges = [(u, v, -w) for (u, v, w) in optimal_arborescence_1]
+
+    G = nx.from_numpy_array(-G_array, create_using=nx.DiGraph)
+
+    # Quickly make sure max branching is empty.
+    x = branchings.maximum_branching(G)
+    x_ = build_branching([])
+    assert_equal_branchings(x, x_)
+
+    # Now test the min branching.
+    x = branchings.minimum_branching(G)
+    x_ = build_branching(edges)
+    assert_equal_branchings(x, x_)
+
+
+def test_edge_attribute_preservation_normal_graph():
+    # Test that edge attributes are preserved when finding an optimum graph
+    # using the Edmonds class for normal graphs.
+    G = nx.Graph()
+
+    edgelist = [
+        (0, 1, [("weight", 5), ("otherattr", 1), ("otherattr2", 3)]),
+        (0, 2, [("weight", 5), ("otherattr", 2), ("otherattr2", 2)]),
+        (1, 2, [("weight", 6), ("otherattr", 3), ("otherattr2", 1)]),
+    ]
+    G.add_edges_from(edgelist)
+
+    B = branchings.maximum_branching(G, preserve_attrs=True)
+
+    assert B[0][1]["otherattr"] == 1
+    assert B[0][1]["otherattr2"] == 3
+
+
+def test_edge_attribute_preservation_multigraph():
+    # Test that edge attributes are preserved when finding an optimum graph
+    # using the Edmonds class for multigraphs.
+    G = nx.MultiGraph()
+
+    edgelist = [
+        (0, 1, [("weight", 5), ("otherattr", 1), ("otherattr2", 3)]),
+        (0, 2, [("weight", 5), ("otherattr", 2), ("otherattr2", 2)]),
+        (1, 2, [("weight", 6), ("otherattr", 3), ("otherattr2", 1)]),
+    ]
+    G.add_edges_from(edgelist * 2)  # Make sure we have duplicate edge paths
+
+    B = branchings.maximum_branching(G, preserve_attrs=True)
+
+    assert B[0][1][0]["otherattr"] == 1
+    assert B[0][1][0]["otherattr2"] == 3
+
+
+def test_edge_attribute_discard():
+    # Test that edge attributes are discarded if we do not specify to keep them
+    G = nx.Graph()
+
+    edgelist = [
+        (0, 1, [("weight", 5), ("otherattr", 1), ("otherattr2", 3)]),
+        (0, 2, [("weight", 5), ("otherattr", 2), ("otherattr2", 2)]),
+        (1, 2, [("weight", 6), ("otherattr", 3), ("otherattr2", 1)]),
+    ]
+    G.add_edges_from(edgelist)
+
+    B = branchings.maximum_branching(G, preserve_attrs=False)
+
+    edge_dict = B[0][1]
+    with pytest.raises(KeyError):
+        _ = edge_dict["otherattr"]
+
+
+def test_partition_spanning_arborescence():
+    """
+    Test that we can generate minimum spanning arborescences which respect the
+    given partition.
+    """
+    G = nx.from_numpy_array(G_array, create_using=nx.DiGraph)
+    G[3][0]["partition"] = nx.EdgePartition.EXCLUDED
+    G[2][3]["partition"] = nx.EdgePartition.INCLUDED
+    G[7][3]["partition"] = nx.EdgePartition.EXCLUDED
+    G[0][2]["partition"] = nx.EdgePartition.EXCLUDED
+    G[6][2]["partition"] = nx.EdgePartition.INCLUDED
+
+    actual_edges = [
+        (0, 4, 12),
+        (1, 0, 4),
+        (1, 5, 13),
+        (2, 3, 21),
+        (4, 7, 12),
+        (5, 6, 14),
+        (5, 8, 12),
+        (6, 2, 21),
+    ]
+
+    B = branchings.minimum_spanning_arborescence(G, partition="partition")
+    assert_equal_branchings(build_branching(actual_edges), B)
+
+
+def test_arborescence_iterator_min():
+    """
+    Tests the arborescence iterator.
+
+    A brute force method found 680 arborescences in this graph.
+    This test will not verify all of them individually, but will check two
+    things
+
+    * The iterator returns 680 arborescences
+    * The weight of the arborescences is non-strictly increasing
+
+    for more information please visit
+    https://mjschwenne.github.io/2021/06/10/implementing-the-iterators.html
+    """
+    G = nx.from_numpy_array(G_array, create_using=nx.DiGraph)
+
+    arborescence_count = 0
+    arborescence_weight = -math.inf
+    for B in branchings.ArborescenceIterator(G):
+        arborescence_count += 1
+        new_arborescence_weight = B.size(weight="weight")
+        assert new_arborescence_weight >= arborescence_weight
+        arborescence_weight = new_arborescence_weight
+
+    assert arborescence_count == 680
+
+
+def test_arborescence_iterator_max():
+    """
+    Tests the arborescence iterator.
+
+    A brute force method found 680 arborescences in this graph.
+    This test will not verify all of them individually, but will check two
+    things
+
+    * The iterator returns 680 arborescences
+    * The weight of the arborescences is non-strictly decreasing
+
+    for more information please visit
+    https://mjschwenne.github.io/2021/06/10/implementing-the-iterators.html
+    """
+    G = nx.from_numpy_array(G_array, create_using=nx.DiGraph)
+
+    arborescence_count = 0
+    arborescence_weight = math.inf
+    for B in branchings.ArborescenceIterator(G, minimum=False):
+        arborescence_count += 1
+        new_arborescence_weight = B.size(weight="weight")
+        assert new_arborescence_weight <= arborescence_weight
+        arborescence_weight = new_arborescence_weight
+
+    assert arborescence_count == 680
+
+
+def test_arborescence_iterator_initial_partition():
+    """
+    Tests the arborescence iterator with three included edges and three excluded
+    in the initial partition.
+
+    A brute force method similar to the one used in the above tests found that
+    there are 16 arborescences which contain the included edges and not the
+    excluded edges.
+    """
+    G = nx.from_numpy_array(G_array, create_using=nx.DiGraph)
+    included_edges = [(1, 0), (5, 6), (8, 7)]
+    excluded_edges = [(0, 2), (3, 6), (1, 5)]
+
+    arborescence_count = 0
+    arborescence_weight = -math.inf
+    for B in branchings.ArborescenceIterator(
+        G, init_partition=(included_edges, excluded_edges)
+    ):
+        arborescence_count += 1
+        new_arborescence_weight = B.size(weight="weight")
+        assert new_arborescence_weight >= arborescence_weight
+        arborescence_weight = new_arborescence_weight
+        for e in included_edges:
+            assert e in B.edges
+        for e in excluded_edges:
+            assert e not in B.edges
+    assert arborescence_count == 16
+
+
+def test_branchings_with_default_weights():
+    """
+    Tests that various branching algorithms work on graphs without weights.
+    For more information, see issue #7279.
+    """
+    graph = nx.erdos_renyi_graph(10, p=0.2, directed=True, seed=123)
+
+    assert all(
+        "weight" not in d for (u, v, d) in graph.edges(data=True)
+    ), "test is for graphs without a weight attribute"
+
+    # Calling these functions will modify graph inplace to add weights
+    # copy the graph to avoid this.
+    nx.minimum_spanning_arborescence(graph.copy())
+    nx.maximum_spanning_arborescence(graph.copy())
+    nx.minimum_branching(graph.copy())
+    nx.maximum_branching(graph.copy())
+    nx.algorithms.tree.minimal_branching(graph.copy())
+    nx.algorithms.tree.branching_weight(graph.copy())
+    nx.algorithms.tree.greedy_branching(graph.copy())
+
+    assert all(
+        "weight" not in d for (u, v, d) in graph.edges(data=True)
+    ), "The above calls should not modify the initial graph in-place"
diff --git a/.venv/lib/python3.12/site-packages/networkx/algorithms/tree/tests/test_coding.py b/.venv/lib/python3.12/site-packages/networkx/algorithms/tree/tests/test_coding.py
new file mode 100644
index 00000000..26bd4083
--- /dev/null
+++ b/.venv/lib/python3.12/site-packages/networkx/algorithms/tree/tests/test_coding.py
@@ -0,0 +1,114 @@
+"""Unit tests for the :mod:`~networkx.algorithms.tree.coding` module."""
+
+from itertools import product
+
+import pytest
+
+import networkx as nx
+from networkx.utils import edges_equal, nodes_equal
+
+
+class TestPruferSequence:
+    """Unit tests for the Prüfer sequence encoding and decoding
+    functions.
+
+    """
+
+    def test_nontree(self):
+        with pytest.raises(nx.NotATree):
+            G = nx.cycle_graph(3)
+            nx.to_prufer_sequence(G)
+
+    def test_null_graph(self):
+        with pytest.raises(nx.NetworkXPointlessConcept):
+            nx.to_prufer_sequence(nx.null_graph())
+
+    def test_trivial_graph(self):
+        with pytest.raises(nx.NetworkXPointlessConcept):
+            nx.to_prufer_sequence(nx.trivial_graph())
+
+    def test_bad_integer_labels(self):
+        with pytest.raises(KeyError):
+            T = nx.Graph(nx.utils.pairwise("abc"))
+            nx.to_prufer_sequence(T)
+
+    def test_encoding(self):
+        """Tests for encoding a tree as a Prüfer sequence using the
+        iterative strategy.
+
+        """
+        # Example from Wikipedia.
+        tree = nx.Graph([(0, 3), (1, 3), (2, 3), (3, 4), (4, 5)])
+        sequence = nx.to_prufer_sequence(tree)
+        assert sequence == [3, 3, 3, 4]
+
+    def test_decoding(self):
+        """Tests for decoding a tree from a Prüfer sequence."""
+        # Example from Wikipedia.
+        sequence = [3, 3, 3, 4]
+        tree = nx.from_prufer_sequence(sequence)
+        assert nodes_equal(list(tree), list(range(6)))
+        edges = [(0, 3), (1, 3), (2, 3), (3, 4), (4, 5)]
+        assert edges_equal(list(tree.edges()), edges)
+
+    def test_decoding2(self):
+        # Example from "An Optimal Algorithm for Prufer Codes".
+        sequence = [2, 4, 0, 1, 3, 3]
+        tree = nx.from_prufer_sequence(sequence)
+        assert nodes_equal(list(tree), list(range(8)))
+        edges = [(0, 1), (0, 4), (1, 3), (2, 4), (2, 5), (3, 6), (3, 7)]
+        assert edges_equal(list(tree.edges()), edges)
+
+    def test_inverse(self):
+        """Tests that the encoding and decoding functions are inverses."""
+        for T in nx.nonisomorphic_trees(4):
+            T2 = nx.from_prufer_sequence(nx.to_prufer_sequence(T))
+            assert nodes_equal(list(T), list(T2))
+            assert edges_equal(list(T.edges()), list(T2.edges()))
+
+        for seq in product(range(4), repeat=2):
+            seq2 = nx.to_prufer_sequence(nx.from_prufer_sequence(seq))
+            assert list(seq) == seq2
+
+
+class TestNestedTuple:
+    """Unit tests for the nested tuple encoding and decoding functions."""
+
+    def test_nontree(self):
+        with pytest.raises(nx.NotATree):
+            G = nx.cycle_graph(3)
+            nx.to_nested_tuple(G, 0)
+
+    def test_unknown_root(self):
+        with pytest.raises(nx.NodeNotFound):
+            G = nx.path_graph(2)
+            nx.to_nested_tuple(G, "bogus")
+
+    def test_encoding(self):
+        T = nx.full_rary_tree(2, 2**3 - 1)
+        expected = (((), ()), ((), ()))
+        actual = nx.to_nested_tuple(T, 0)
+        assert nodes_equal(expected, actual)
+
+    def test_canonical_form(self):
+        T = nx.Graph()
+        T.add_edges_from([(0, 1), (0, 2), (0, 3)])
+        T.add_edges_from([(1, 4), (1, 5)])
+        T.add_edges_from([(3, 6), (3, 7)])
+        root = 0
+        actual = nx.to_nested_tuple(T, root, canonical_form=True)
+        expected = ((), ((), ()), ((), ()))
+        assert actual == expected
+
+    def test_decoding(self):
+        balanced = (((), ()), ((), ()))
+        expected = nx.full_rary_tree(2, 2**3 - 1)
+        actual = nx.from_nested_tuple(balanced)
+        assert nx.is_isomorphic(expected, actual)
+
+    def test_sensible_relabeling(self):
+        balanced = (((), ()), ((), ()))
+        T = nx.from_nested_tuple(balanced, sensible_relabeling=True)
+        edges = [(0, 1), (0, 2), (1, 3), (1, 4), (2, 5), (2, 6)]
+        assert nodes_equal(list(T), list(range(2**3 - 1)))
+        assert edges_equal(list(T.edges()), edges)
diff --git a/.venv/lib/python3.12/site-packages/networkx/algorithms/tree/tests/test_decomposition.py b/.venv/lib/python3.12/site-packages/networkx/algorithms/tree/tests/test_decomposition.py
new file mode 100644
index 00000000..8c376053
--- /dev/null
+++ b/.venv/lib/python3.12/site-packages/networkx/algorithms/tree/tests/test_decomposition.py
@@ -0,0 +1,79 @@
+import networkx as nx
+from networkx.algorithms.tree.decomposition import junction_tree
+
+
+def test_junction_tree_directed_confounders():
+    B = nx.DiGraph()
+    B.add_edges_from([("A", "C"), ("B", "C"), ("C", "D"), ("C", "E")])
+
+    G = junction_tree(B)
+    J = nx.Graph()
+    J.add_edges_from(
+        [
+            (("C", "E"), ("C",)),
+            (("C",), ("A", "B", "C")),
+            (("A", "B", "C"), ("C",)),
+            (("C",), ("C", "D")),
+        ]
+    )
+
+    assert nx.is_isomorphic(G, J)
+
+
+def test_junction_tree_directed_unconnected_nodes():
+    B = nx.DiGraph()
+    B.add_nodes_from([("A", "B", "C", "D")])
+    G = junction_tree(B)
+
+    J = nx.Graph()
+    J.add_nodes_from([("A", "B", "C", "D")])
+
+    assert nx.is_isomorphic(G, J)
+
+
+def test_junction_tree_directed_cascade():
+    B = nx.DiGraph()
+    B.add_edges_from([("A", "B"), ("B", "C"), ("C", "D")])
+    G = junction_tree(B)
+
+    J = nx.Graph()
+    J.add_edges_from(
+        [
+            (("A", "B"), ("B",)),
+            (("B",), ("B", "C")),
+            (("B", "C"), ("C",)),
+            (("C",), ("C", "D")),
+        ]
+    )
+    assert nx.is_isomorphic(G, J)
+
+
+def test_junction_tree_directed_unconnected_edges():
+    B = nx.DiGraph()
+    B.add_edges_from([("A", "B"), ("C", "D"), ("E", "F")])
+    G = junction_tree(B)
+
+    J = nx.Graph()
+    J.add_nodes_from([("A", "B"), ("C", "D"), ("E", "F")])
+
+    assert nx.is_isomorphic(G, J)
+
+
+def test_junction_tree_undirected():
+    B = nx.Graph()
+    B.add_edges_from([("A", "C"), ("A", "D"), ("B", "C"), ("C", "E")])
+    G = junction_tree(B)
+
+    J = nx.Graph()
+    J.add_edges_from(
+        [
+            (("A", "D"), ("A",)),
+            (("A",), ("A", "C")),
+            (("A", "C"), ("C",)),
+            (("C",), ("B", "C")),
+            (("B", "C"), ("C",)),
+            (("C",), ("C", "E")),
+        ]
+    )
+
+    assert nx.is_isomorphic(G, J)
diff --git a/.venv/lib/python3.12/site-packages/networkx/algorithms/tree/tests/test_mst.py b/.venv/lib/python3.12/site-packages/networkx/algorithms/tree/tests/test_mst.py
new file mode 100644
index 00000000..f8945a71
--- /dev/null
+++ b/.venv/lib/python3.12/site-packages/networkx/algorithms/tree/tests/test_mst.py
@@ -0,0 +1,918 @@
+"""Unit tests for the :mod:`networkx.algorithms.tree.mst` module."""
+
+import pytest
+
+import networkx as nx
+from networkx.utils import edges_equal, nodes_equal
+
+
+def test_unknown_algorithm():
+    with pytest.raises(ValueError):
+        nx.minimum_spanning_tree(nx.Graph(), algorithm="random")
+    with pytest.raises(
+        ValueError, match="random is not a valid choice for an algorithm."
+    ):
+        nx.maximum_spanning_edges(nx.Graph(), algorithm="random")
+
+
+class MinimumSpanningTreeTestBase:
+    """Base class for test classes for minimum spanning tree algorithms.
+    This class contains some common tests that will be inherited by
+    subclasses. Each subclass must have a class attribute
+    :data:`algorithm` that is a string representing the algorithm to
+    run, as described under the ``algorithm`` keyword argument for the
+    :func:`networkx.minimum_spanning_edges` function.  Subclasses can
+    then implement any algorithm-specific tests.
+    """
+
+    def setup_method(self, method):
+        """Creates an example graph and stores the expected minimum and
+        maximum spanning tree edges.
+        """
+        # This stores the class attribute `algorithm` in an instance attribute.
+        self.algo = self.algorithm
+        # This example graph comes from Wikipedia:
+        # https://en.wikipedia.org/wiki/Kruskal's_algorithm
+        edges = [
+            (0, 1, 7),
+            (0, 3, 5),
+            (1, 2, 8),
+            (1, 3, 9),
+            (1, 4, 7),
+            (2, 4, 5),
+            (3, 4, 15),
+            (3, 5, 6),
+            (4, 5, 8),
+            (4, 6, 9),
+            (5, 6, 11),
+        ]
+        self.G = nx.Graph()
+        self.G.add_weighted_edges_from(edges)
+        self.minimum_spanning_edgelist = [
+            (0, 1, {"weight": 7}),
+            (0, 3, {"weight": 5}),
+            (1, 4, {"weight": 7}),
+            (2, 4, {"weight": 5}),
+            (3, 5, {"weight": 6}),
+            (4, 6, {"weight": 9}),
+        ]
+        self.maximum_spanning_edgelist = [
+            (0, 1, {"weight": 7}),
+            (1, 2, {"weight": 8}),
+            (1, 3, {"weight": 9}),
+            (3, 4, {"weight": 15}),
+            (4, 6, {"weight": 9}),
+            (5, 6, {"weight": 11}),
+        ]
+
+    def test_minimum_edges(self):
+        edges = nx.minimum_spanning_edges(self.G, algorithm=self.algo)
+        # Edges from the spanning edges functions don't come in sorted
+        # orientation, so we need to sort each edge individually.
+        actual = sorted((min(u, v), max(u, v), d) for u, v, d in edges)
+        assert edges_equal(actual, self.minimum_spanning_edgelist)
+
+    def test_maximum_edges(self):
+        edges = nx.maximum_spanning_edges(self.G, algorithm=self.algo)
+        # Edges from the spanning edges functions don't come in sorted
+        # orientation, so we need to sort each edge individually.
+        actual = sorted((min(u, v), max(u, v), d) for u, v, d in edges)
+        assert edges_equal(actual, self.maximum_spanning_edgelist)
+
+    def test_without_data(self):
+        edges = nx.minimum_spanning_edges(self.G, algorithm=self.algo, data=False)
+        # Edges from the spanning edges functions don't come in sorted
+        # orientation, so we need to sort each edge individually.
+        actual = sorted((min(u, v), max(u, v)) for u, v in edges)
+        expected = [(u, v) for u, v, d in self.minimum_spanning_edgelist]
+        assert edges_equal(actual, expected)
+
+    def test_nan_weights(self):
+        # Edge weights NaN never appear in the spanning tree. see #2164
+        G = self.G
+        G.add_edge(0, 12, weight=float("nan"))
+        edges = nx.minimum_spanning_edges(
+            G, algorithm=self.algo, data=False, ignore_nan=True
+        )
+        actual = sorted((min(u, v), max(u, v)) for u, v in edges)
+        expected = [(u, v) for u, v, d in self.minimum_spanning_edgelist]
+        assert edges_equal(actual, expected)
+        # Now test for raising exception
+        edges = nx.minimum_spanning_edges(
+            G, algorithm=self.algo, data=False, ignore_nan=False
+        )
+        with pytest.raises(ValueError):
+            list(edges)
+        # test default for ignore_nan as False
+        edges = nx.minimum_spanning_edges(G, algorithm=self.algo, data=False)
+        with pytest.raises(ValueError):
+            list(edges)
+
+    def test_nan_weights_MultiGraph(self):
+        G = nx.MultiGraph()
+        G.add_edge(0, 12, weight=float("nan"))
+        edges = nx.minimum_spanning_edges(
+            G, algorithm="prim", data=False, ignore_nan=False
+        )
+        with pytest.raises(ValueError):
+            list(edges)
+        # test default for ignore_nan as False
+        edges = nx.minimum_spanning_edges(G, algorithm="prim", data=False)
+        with pytest.raises(ValueError):
+            list(edges)
+
+    def test_nan_weights_order(self):
+        # now try again with a nan edge at the beginning of G.nodes
+        edges = [
+            (0, 1, 7),
+            (0, 3, 5),
+            (1, 2, 8),
+            (1, 3, 9),
+            (1, 4, 7),
+            (2, 4, 5),
+            (3, 4, 15),
+            (3, 5, 6),
+            (4, 5, 8),
+            (4, 6, 9),
+            (5, 6, 11),
+        ]
+        G = nx.Graph()
+        G.add_weighted_edges_from([(u + 1, v + 1, wt) for u, v, wt in edges])
+        G.add_edge(0, 7, weight=float("nan"))
+        edges = nx.minimum_spanning_edges(
+            G, algorithm=self.algo, data=False, ignore_nan=True
+        )
+        actual = sorted((min(u, v), max(u, v)) for u, v in edges)
+        shift = [(u + 1, v + 1) for u, v, d in self.minimum_spanning_edgelist]
+        assert edges_equal(actual, shift)
+
+    def test_isolated_node(self):
+        # now try again with an isolated node
+        edges = [
+            (0, 1, 7),
+            (0, 3, 5),
+            (1, 2, 8),
+            (1, 3, 9),
+            (1, 4, 7),
+            (2, 4, 5),
+            (3, 4, 15),
+            (3, 5, 6),
+            (4, 5, 8),
+            (4, 6, 9),
+            (5, 6, 11),
+        ]
+        G = nx.Graph()
+        G.add_weighted_edges_from([(u + 1, v + 1, wt) for u, v, wt in edges])
+        G.add_node(0)
+        edges = nx.minimum_spanning_edges(
+            G, algorithm=self.algo, data=False, ignore_nan=True
+        )
+        actual = sorted((min(u, v), max(u, v)) for u, v in edges)
+        shift = [(u + 1, v + 1) for u, v, d in self.minimum_spanning_edgelist]
+        assert edges_equal(actual, shift)
+
+    def test_minimum_tree(self):
+        T = nx.minimum_spanning_tree(self.G, algorithm=self.algo)
+        actual = sorted(T.edges(data=True))
+        assert edges_equal(actual, self.minimum_spanning_edgelist)
+
+    def test_maximum_tree(self):
+        T = nx.maximum_spanning_tree(self.G, algorithm=self.algo)
+        actual = sorted(T.edges(data=True))
+        assert edges_equal(actual, self.maximum_spanning_edgelist)
+
+    def test_disconnected(self):
+        G = nx.Graph([(0, 1, {"weight": 1}), (2, 3, {"weight": 2})])
+        T = nx.minimum_spanning_tree(G, algorithm=self.algo)
+        assert nodes_equal(list(T), list(range(4)))
+        assert edges_equal(list(T.edges()), [(0, 1), (2, 3)])
+
+    def test_empty_graph(self):
+        G = nx.empty_graph(3)
+        T = nx.minimum_spanning_tree(G, algorithm=self.algo)
+        assert nodes_equal(sorted(T), list(range(3)))
+        assert T.number_of_edges() == 0
+
+    def test_attributes(self):
+        G = nx.Graph()
+        G.add_edge(1, 2, weight=1, color="red", distance=7)
+        G.add_edge(2, 3, weight=1, color="green", distance=2)
+        G.add_edge(1, 3, weight=10, color="blue", distance=1)
+        G.graph["foo"] = "bar"
+        T = nx.minimum_spanning_tree(G, algorithm=self.algo)
+        assert T.graph == G.graph
+        assert nodes_equal(T, G)
+        for u, v in T.edges():
+            assert T.adj[u][v] == G.adj[u][v]
+
+    def test_weight_attribute(self):
+        G = nx.Graph()
+        G.add_edge(0, 1, weight=1, distance=7)
+        G.add_edge(0, 2, weight=30, distance=1)
+        G.add_edge(1, 2, weight=1, distance=1)
+        G.add_node(3)
+        T = nx.minimum_spanning_tree(G, algorithm=self.algo, weight="distance")
+        assert nodes_equal(sorted(T), list(range(4)))
+        assert edges_equal(sorted(T.edges()), [(0, 2), (1, 2)])
+        T = nx.maximum_spanning_tree(G, algorithm=self.algo, weight="distance")
+        assert nodes_equal(sorted(T), list(range(4)))
+        assert edges_equal(sorted(T.edges()), [(0, 1), (0, 2)])
+
+
+class TestBoruvka(MinimumSpanningTreeTestBase):
+    """Unit tests for computing a minimum (or maximum) spanning tree
+    using Borůvka's algorithm.
+    """
+
+    algorithm = "boruvka"
+
+    def test_unicode_name(self):
+        """Tests that using a Unicode string can correctly indicate
+        Borůvka's algorithm.
+        """
+        edges = nx.minimum_spanning_edges(self.G, algorithm="borůvka")
+        # Edges from the spanning edges functions don't come in sorted
+        # orientation, so we need to sort each edge individually.
+        actual = sorted((min(u, v), max(u, v), d) for u, v, d in edges)
+        assert edges_equal(actual, self.minimum_spanning_edgelist)
+
+
+class MultigraphMSTTestBase(MinimumSpanningTreeTestBase):
+    # Abstract class
+
+    def test_multigraph_keys_min(self):
+        """Tests that the minimum spanning edges of a multigraph
+        preserves edge keys.
+        """
+        G = nx.MultiGraph()
+        G.add_edge(0, 1, key="a", weight=2)
+        G.add_edge(0, 1, key="b", weight=1)
+        min_edges = nx.minimum_spanning_edges
+        mst_edges = min_edges(G, algorithm=self.algo, data=False)
+        assert edges_equal([(0, 1, "b")], list(mst_edges))
+
+    def test_multigraph_keys_max(self):
+        """Tests that the maximum spanning edges of a multigraph
+        preserves edge keys.
+        """
+        G = nx.MultiGraph()
+        G.add_edge(0, 1, key="a", weight=2)
+        G.add_edge(0, 1, key="b", weight=1)
+        max_edges = nx.maximum_spanning_edges
+        mst_edges = max_edges(G, algorithm=self.algo, data=False)
+        assert edges_equal([(0, 1, "a")], list(mst_edges))
+
+
+class TestKruskal(MultigraphMSTTestBase):
+    """Unit tests for computing a minimum (or maximum) spanning tree
+    using Kruskal's algorithm.
+    """
+
+    algorithm = "kruskal"
+
+    def test_key_data_bool(self):
+        """Tests that the keys and data values are included in
+        MST edges based on whether keys and data parameters are
+        true or false"""
+        G = nx.MultiGraph()
+        G.add_edge(1, 2, key=1, weight=2)
+        G.add_edge(1, 2, key=2, weight=3)
+        G.add_edge(3, 2, key=1, weight=2)
+        G.add_edge(3, 1, key=1, weight=4)
+
+        # keys are included and data is not included
+        mst_edges = nx.minimum_spanning_edges(
+            G, algorithm=self.algo, keys=True, data=False
+        )
+        assert edges_equal([(1, 2, 1), (2, 3, 1)], list(mst_edges))
+
+        # keys are not included and data is included
+        mst_edges = nx.minimum_spanning_edges(
+            G, algorithm=self.algo, keys=False, data=True
+        )
+        assert edges_equal(
+            [(1, 2, {"weight": 2}), (2, 3, {"weight": 2})], list(mst_edges)
+        )
+
+        # both keys and data are not included
+        mst_edges = nx.minimum_spanning_edges(
+            G, algorithm=self.algo, keys=False, data=False
+        )
+        assert edges_equal([(1, 2), (2, 3)], list(mst_edges))
+
+        # both keys and data are included
+        mst_edges = nx.minimum_spanning_edges(
+            G, algorithm=self.algo, keys=True, data=True
+        )
+        assert edges_equal(
+            [(1, 2, 1, {"weight": 2}), (2, 3, 1, {"weight": 2})], list(mst_edges)
+        )
+
+
+class TestPrim(MultigraphMSTTestBase):
+    """Unit tests for computing a minimum (or maximum) spanning tree
+    using Prim's algorithm.
+    """
+
+    algorithm = "prim"
+
+    def test_prim_mst_edges_simple_graph(self):
+        H = nx.Graph()
+        H.add_edge(1, 2, key=2, weight=3)
+        H.add_edge(3, 2, key=1, weight=2)
+        H.add_edge(3, 1, key=1, weight=4)
+
+        mst_edges = nx.minimum_spanning_edges(H, algorithm=self.algo, ignore_nan=True)
+        assert edges_equal(
+            [(1, 2, {"key": 2, "weight": 3}), (2, 3, {"key": 1, "weight": 2})],
+            list(mst_edges),
+        )
+
+    def test_ignore_nan(self):
+        """Tests that the edges with NaN weights are ignored or
+        raise an Error based on ignore_nan is true or false"""
+        H = nx.MultiGraph()
+        H.add_edge(1, 2, key=1, weight=float("nan"))
+        H.add_edge(1, 2, key=2, weight=3)
+        H.add_edge(3, 2, key=1, weight=2)
+        H.add_edge(3, 1, key=1, weight=4)
+
+        # NaN weight edges are ignored when ignore_nan=True
+        mst_edges = nx.minimum_spanning_edges(H, algorithm=self.algo, ignore_nan=True)
+        assert edges_equal(
+            [(1, 2, 2, {"weight": 3}), (2, 3, 1, {"weight": 2})], list(mst_edges)
+        )
+
+        # NaN weight edges raise Error when ignore_nan=False
+        with pytest.raises(ValueError):
+            list(nx.minimum_spanning_edges(H, algorithm=self.algo, ignore_nan=False))
+
+    def test_multigraph_keys_tree(self):
+        G = nx.MultiGraph()
+        G.add_edge(0, 1, key="a", weight=2)
+        G.add_edge(0, 1, key="b", weight=1)
+        T = nx.minimum_spanning_tree(G, algorithm=self.algo)
+        assert edges_equal([(0, 1, 1)], list(T.edges(data="weight")))
+
+    def test_multigraph_keys_tree_max(self):
+        G = nx.MultiGraph()
+        G.add_edge(0, 1, key="a", weight=2)
+        G.add_edge(0, 1, key="b", weight=1)
+        T = nx.maximum_spanning_tree(G, algorithm=self.algo)
+        assert edges_equal([(0, 1, 2)], list(T.edges(data="weight")))
+
+
+class TestSpanningTreeIterator:
+    """
+    Tests the spanning tree iterator on the example graph in the 2005 Sörensen
+    and Janssens paper An Algorithm to Generate all Spanning Trees of a Graph in
+    Order of Increasing Cost
+    """
+
+    def setup_method(self):
+        # Original Graph
+        edges = [(0, 1, 5), (1, 2, 4), (1, 4, 6), (2, 3, 5), (2, 4, 7), (3, 4, 3)]
+        self.G = nx.Graph()
+        self.G.add_weighted_edges_from(edges)
+        # List of lists of spanning trees in increasing order
+        self.spanning_trees = [
+            # 1, MST, cost = 17
+            [
+                (0, 1, {"weight": 5}),
+                (1, 2, {"weight": 4}),
+                (2, 3, {"weight": 5}),
+                (3, 4, {"weight": 3}),
+            ],
+            # 2, cost = 18
+            [
+                (0, 1, {"weight": 5}),
+                (1, 2, {"weight": 4}),
+                (1, 4, {"weight": 6}),
+                (3, 4, {"weight": 3}),
+            ],
+            # 3, cost = 19
+            [
+                (0, 1, {"weight": 5}),
+                (1, 4, {"weight": 6}),
+                (2, 3, {"weight": 5}),
+                (3, 4, {"weight": 3}),
+            ],
+            # 4, cost = 19
+            [
+                (0, 1, {"weight": 5}),
+                (1, 2, {"weight": 4}),
+                (2, 4, {"weight": 7}),
+                (3, 4, {"weight": 3}),
+            ],
+            # 5, cost = 20
+            [
+                (0, 1, {"weight": 5}),
+                (1, 2, {"weight": 4}),
+                (1, 4, {"weight": 6}),
+                (2, 3, {"weight": 5}),
+            ],
+            # 6, cost = 21
+            [
+                (0, 1, {"weight": 5}),
+                (1, 4, {"weight": 6}),
+                (2, 4, {"weight": 7}),
+                (3, 4, {"weight": 3}),
+            ],
+            # 7, cost = 21
+            [
+                (0, 1, {"weight": 5}),
+                (1, 2, {"weight": 4}),
+                (2, 3, {"weight": 5}),
+                (2, 4, {"weight": 7}),
+            ],
+            # 8, cost = 23
+            [
+                (0, 1, {"weight": 5}),
+                (1, 4, {"weight": 6}),
+                (2, 3, {"weight": 5}),
+                (2, 4, {"weight": 7}),
+            ],
+        ]
+
+    def test_minimum_spanning_tree_iterator(self):
+        """
+        Tests that the spanning trees are correctly returned in increasing order
+        """
+        tree_index = 0
+        for tree in nx.SpanningTreeIterator(self.G):
+            actual = sorted(tree.edges(data=True))
+            assert edges_equal(actual, self.spanning_trees[tree_index])
+            tree_index += 1
+
+    def test_maximum_spanning_tree_iterator(self):
+        """
+        Tests that the spanning trees are correctly returned in decreasing order
+        """
+        tree_index = 7
+        for tree in nx.SpanningTreeIterator(self.G, minimum=False):
+            actual = sorted(tree.edges(data=True))
+            assert edges_equal(actual, self.spanning_trees[tree_index])
+            tree_index -= 1
+
+
+class TestSpanningTreeMultiGraphIterator:
+    """
+    Uses the same graph as the above class but with an added edge of twice the weight.
+    """
+
+    def setup_method(self):
+        # New graph
+        edges = [
+            (0, 1, 5),
+            (0, 1, 10),
+            (1, 2, 4),
+            (1, 2, 8),
+            (1, 4, 6),
+            (1, 4, 12),
+            (2, 3, 5),
+            (2, 3, 10),
+            (2, 4, 7),
+            (2, 4, 14),
+            (3, 4, 3),
+            (3, 4, 6),
+        ]
+        self.G = nx.MultiGraph()
+        self.G.add_weighted_edges_from(edges)
+
+        # There are 128 trees. I'd rather not list all 128 here, and computing them
+        # on such a small graph actually doesn't take that long.
+        from itertools import combinations
+
+        self.spanning_trees = []
+        for e in combinations(self.G.edges, 4):
+            tree = self.G.edge_subgraph(e)
+            if nx.is_tree(tree):
+                self.spanning_trees.append(sorted(tree.edges(keys=True, data=True)))
+
+    def test_minimum_spanning_tree_iterator_multigraph(self):
+        """
+        Tests that the spanning trees are correctly returned in increasing order
+        """
+        tree_index = 0
+        last_weight = 0
+        for tree in nx.SpanningTreeIterator(self.G):
+            actual = sorted(tree.edges(keys=True, data=True))
+            weight = sum([e[3]["weight"] for e in actual])
+            assert actual in self.spanning_trees
+            assert weight >= last_weight
+            tree_index += 1
+
+    def test_maximum_spanning_tree_iterator_multigraph(self):
+        """
+        Tests that the spanning trees are correctly returned in decreasing order
+        """
+        tree_index = 127
+        # Maximum weight tree is 46
+        last_weight = 50
+        for tree in nx.SpanningTreeIterator(self.G, minimum=False):
+            actual = sorted(tree.edges(keys=True, data=True))
+            weight = sum([e[3]["weight"] for e in actual])
+            assert actual in self.spanning_trees
+            assert weight <= last_weight
+            tree_index -= 1
+
+
+def test_random_spanning_tree_multiplicative_small():
+    """
+    Using a fixed seed, sample one tree for repeatability.
+    """
+    from math import exp
+
+    pytest.importorskip("scipy")
+
+    gamma = {
+        (0, 1): -0.6383,
+        (0, 2): -0.6827,
+        (0, 5): 0,
+        (1, 2): -1.0781,
+        (1, 4): 0,
+        (2, 3): 0,
+        (5, 3): -0.2820,
+        (5, 4): -0.3327,
+        (4, 3): -0.9927,
+    }
+
+    # The undirected support of gamma
+    G = nx.Graph()
+    for u, v in gamma:
+        G.add_edge(u, v, lambda_key=exp(gamma[(u, v)]))
+
+    solution_edges = [(2, 3), (3, 4), (0, 5), (5, 4), (4, 1)]
+    solution = nx.Graph()
+    solution.add_edges_from(solution_edges)
+
+    sampled_tree = nx.random_spanning_tree(G, "lambda_key", seed=42)
+
+    assert nx.utils.edges_equal(solution.edges, sampled_tree.edges)
+
+
+@pytest.mark.slow
+def test_random_spanning_tree_multiplicative_large():
+    """
+    Sample many trees from the distribution created in the last test
+    """
+    from math import exp
+    from random import Random
+
+    pytest.importorskip("numpy")
+    stats = pytest.importorskip("scipy.stats")
+
+    gamma = {
+        (0, 1): -0.6383,
+        (0, 2): -0.6827,
+        (0, 5): 0,
+        (1, 2): -1.0781,
+        (1, 4): 0,
+        (2, 3): 0,
+        (5, 3): -0.2820,
+        (5, 4): -0.3327,
+        (4, 3): -0.9927,
+    }
+
+    # The undirected support of gamma
+    G = nx.Graph()
+    for u, v in gamma:
+        G.add_edge(u, v, lambda_key=exp(gamma[(u, v)]))
+
+    # Find the multiplicative weight for each tree.
+    total_weight = 0
+    tree_expected = {}
+    for t in nx.SpanningTreeIterator(G):
+        # Find the multiplicative weight of the spanning tree
+        weight = 1
+        for u, v, d in t.edges(data="lambda_key"):
+            weight *= d
+        tree_expected[t] = weight
+        total_weight += weight
+
+    # Assert that every tree has an entry in the expected distribution
+    assert len(tree_expected) == 75
+
+    # Set the sample size and then calculate the expected number of times we
+    # expect to see each tree. This test uses a near minimum sample size where
+    # the most unlikely tree has an expected frequency of 5.15.
+    # (Minimum required is 5)
+    #
+    # Here we also initialize the tree_actual dict so that we know the keys
+    # match between the two. We will later take advantage of the fact that since
+    # python 3.7 dict order is guaranteed so the expected and actual data will
+    # have the same order.
+    sample_size = 1200
+    tree_actual = {}
+    for t in tree_expected:
+        tree_expected[t] = (tree_expected[t] / total_weight) * sample_size
+        tree_actual[t] = 0
+
+    # Sample the spanning trees
+    #
+    # Assert that they are actually trees and record which of the 75 trees we
+    # have sampled.
+    #
+    # For repeatability, we want to take advantage of the decorators in NetworkX
+    # to randomly sample the same sample each time. However, if we pass in a
+    # constant seed to sample_spanning_tree we will get the same tree each time.
+    # Instead, we can create our own random number generator with a fixed seed
+    # and pass those into sample_spanning_tree.
+    rng = Random(37)
+    for _ in range(sample_size):
+        sampled_tree = nx.random_spanning_tree(G, "lambda_key", seed=rng)
+        assert nx.is_tree(sampled_tree)
+
+        for t in tree_expected:
+            if nx.utils.edges_equal(t.edges, sampled_tree.edges):
+                tree_actual[t] += 1
+                break
+
+    # Conduct a Chi squared test to see if the actual distribution matches the
+    # expected one at an alpha = 0.05 significance level.
+    #
+    # H_0: The distribution of trees in tree_actual matches the normalized product
+    # of the edge weights in the tree.
+    #
+    # H_a: The distribution of trees in tree_actual follows some other
+    # distribution of spanning trees.
+    _, p = stats.chisquare(list(tree_actual.values()), list(tree_expected.values()))
+
+    # Assert that p is greater than the significance level so that we do not
+    # reject the null hypothesis
+    assert not p < 0.05
+
+
+def test_random_spanning_tree_additive_small():
+    """
+    Sample a single spanning tree from the additive method.
+    """
+    pytest.importorskip("scipy")
+
+    edges = {
+        (0, 1): 1,
+        (0, 2): 1,
+        (0, 5): 3,
+        (1, 2): 2,
+        (1, 4): 3,
+        (2, 3): 3,
+        (5, 3): 4,
+        (5, 4): 5,
+        (4, 3): 4,
+    }
+
+    # Build the graph
+    G = nx.Graph()
+    for u, v in edges:
+        G.add_edge(u, v, weight=edges[(u, v)])
+
+    solution_edges = [(0, 2), (1, 2), (2, 3), (3, 4), (3, 5)]
+    solution = nx.Graph()
+    solution.add_edges_from(solution_edges)
+
+    sampled_tree = nx.random_spanning_tree(
+        G, weight="weight", multiplicative=False, seed=37
+    )
+
+    assert nx.utils.edges_equal(solution.edges, sampled_tree.edges)
+
+
+@pytest.mark.slow
+def test_random_spanning_tree_additive_large():
+    """
+    Sample many spanning trees from the additive method.
+    """
+    from random import Random
+
+    pytest.importorskip("numpy")
+    stats = pytest.importorskip("scipy.stats")
+
+    edges = {
+        (0, 1): 1,
+        (0, 2): 1,
+        (0, 5): 3,
+        (1, 2): 2,
+        (1, 4): 3,
+        (2, 3): 3,
+        (5, 3): 4,
+        (5, 4): 5,
+        (4, 3): 4,
+    }
+
+    # Build the graph
+    G = nx.Graph()
+    for u, v in edges:
+        G.add_edge(u, v, weight=edges[(u, v)])
+
+    # Find the additive weight for each tree.
+    total_weight = 0
+    tree_expected = {}
+    for t in nx.SpanningTreeIterator(G):
+        # Find the multiplicative weight of the spanning tree
+        weight = 0
+        for u, v, d in t.edges(data="weight"):
+            weight += d
+        tree_expected[t] = weight
+        total_weight += weight
+
+    # Assert that every tree has an entry in the expected distribution
+    assert len(tree_expected) == 75
+
+    # Set the sample size and then calculate the expected number of times we
+    # expect to see each tree. This test uses a near minimum sample size where
+    # the most unlikely tree has an expected frequency of 5.07.
+    # (Minimum required is 5)
+    #
+    # Here we also initialize the tree_actual dict so that we know the keys
+    # match between the two. We will later take advantage of the fact that since
+    # python 3.7 dict order is guaranteed so the expected and actual data will
+    # have the same order.
+    sample_size = 500
+    tree_actual = {}
+    for t in tree_expected:
+        tree_expected[t] = (tree_expected[t] / total_weight) * sample_size
+        tree_actual[t] = 0
+
+    # Sample the spanning trees
+    #
+    # Assert that they are actually trees and record which of the 75 trees we
+    # have sampled.
+    #
+    # For repeatability, we want to take advantage of the decorators in NetworkX
+    # to randomly sample the same sample each time. However, if we pass in a
+    # constant seed to sample_spanning_tree we will get the same tree each time.
+    # Instead, we can create our own random number generator with a fixed seed
+    # and pass those into sample_spanning_tree.
+    rng = Random(37)
+    for _ in range(sample_size):
+        sampled_tree = nx.random_spanning_tree(
+            G, "weight", multiplicative=False, seed=rng
+        )
+        assert nx.is_tree(sampled_tree)
+
+        for t in tree_expected:
+            if nx.utils.edges_equal(t.edges, sampled_tree.edges):
+                tree_actual[t] += 1
+                break
+
+    # Conduct a Chi squared test to see if the actual distribution matches the
+    # expected one at an alpha = 0.05 significance level.
+    #
+    # H_0: The distribution of trees in tree_actual matches the normalized product
+    # of the edge weights in the tree.
+    #
+    # H_a: The distribution of trees in tree_actual follows some other
+    # distribution of spanning trees.
+    _, p = stats.chisquare(list(tree_actual.values()), list(tree_expected.values()))
+
+    # Assert that p is greater than the significance level so that we do not
+    # reject the null hypothesis
+    assert not p < 0.05
+
+
+def test_random_spanning_tree_empty_graph():
+    G = nx.Graph()
+    rst = nx.tree.random_spanning_tree(G)
+    assert len(rst.nodes) == 0
+    assert len(rst.edges) == 0
+
+
+def test_random_spanning_tree_single_node_graph():
+    G = nx.Graph()
+    G.add_node(0)
+    rst = nx.tree.random_spanning_tree(G)
+    assert len(rst.nodes) == 1
+    assert len(rst.edges) == 0
+
+
+def test_random_spanning_tree_single_node_loop():
+    G = nx.Graph()
+    G.add_node(0)
+    G.add_edge(0, 0)
+    rst = nx.tree.random_spanning_tree(G)
+    assert len(rst.nodes) == 1
+    assert len(rst.edges) == 0
+
+
+class TestNumberSpanningTrees:
+    @classmethod
+    def setup_class(cls):
+        global np
+        np = pytest.importorskip("numpy")
+        sp = pytest.importorskip("scipy")
+
+    def test_nst_disconnected(self):
+        G = nx.empty_graph(2)
+        assert np.isclose(nx.number_of_spanning_trees(G), 0)
+
+    def test_nst_no_nodes(self):
+        G = nx.Graph()
+        with pytest.raises(nx.NetworkXPointlessConcept):
+            nx.number_of_spanning_trees(G)
+
+    def test_nst_weight(self):
+        G = nx.Graph()
+        G.add_edge(1, 2, weight=1)
+        G.add_edge(1, 3, weight=1)
+        G.add_edge(2, 3, weight=2)
+        # weights are ignored
+        assert np.isclose(nx.number_of_spanning_trees(G), 3)
+        # including weight
+        assert np.isclose(nx.number_of_spanning_trees(G, weight="weight"), 5)
+
+    def test_nst_negative_weight(self):
+        G = nx.Graph()
+        G.add_edge(1, 2, weight=1)
+        G.add_edge(1, 3, weight=-1)
+        G.add_edge(2, 3, weight=-2)
+        # weights are ignored
+        assert np.isclose(nx.number_of_spanning_trees(G), 3)
+        # including weight
+        assert np.isclose(nx.number_of_spanning_trees(G, weight="weight"), -1)
+
+    def test_nst_selfloop(self):
+        # self-loops are ignored
+        G = nx.complete_graph(3)
+        G.add_edge(1, 1)
+        assert np.isclose(nx.number_of_spanning_trees(G), 3)
+
+    def test_nst_multigraph(self):
+        G = nx.MultiGraph()
+        G.add_edge(1, 2)
+        G.add_edge(1, 2)
+        G.add_edge(1, 3)
+        G.add_edge(2, 3)
+        assert np.isclose(nx.number_of_spanning_trees(G), 5)
+
+    def test_nst_complete_graph(self):
+        # this is known as Cayley's formula
+        N = 5
+        G = nx.complete_graph(N)
+        assert np.isclose(nx.number_of_spanning_trees(G), N ** (N - 2))
+
+    def test_nst_path_graph(self):
+        G = nx.path_graph(5)
+        assert np.isclose(nx.number_of_spanning_trees(G), 1)
+
+    def test_nst_cycle_graph(self):
+        G = nx.cycle_graph(5)
+        assert np.isclose(nx.number_of_spanning_trees(G), 5)
+
+    def test_nst_directed_noroot(self):
+        G = nx.empty_graph(3, create_using=nx.MultiDiGraph)
+        with pytest.raises(nx.NetworkXError):
+            nx.number_of_spanning_trees(G)
+
+    def test_nst_directed_root_not_exist(self):
+        G = nx.empty_graph(3, create_using=nx.MultiDiGraph)
+        with pytest.raises(nx.NetworkXError):
+            nx.number_of_spanning_trees(G, root=42)
+
+    def test_nst_directed_not_weak_connected(self):
+        G = nx.DiGraph()
+        G.add_edge(1, 2)
+        G.add_edge(3, 4)
+        assert np.isclose(nx.number_of_spanning_trees(G, root=1), 0)
+
+    def test_nst_directed_cycle_graph(self):
+        G = nx.DiGraph()
+        G = nx.cycle_graph(7, G)
+        assert np.isclose(nx.number_of_spanning_trees(G, root=0), 1)
+
+    def test_nst_directed_complete_graph(self):
+        G = nx.DiGraph()
+        G = nx.complete_graph(7, G)
+        assert np.isclose(nx.number_of_spanning_trees(G, root=0), 7**5)
+
+    def test_nst_directed_multi(self):
+        G = nx.MultiDiGraph()
+        G = nx.cycle_graph(3, G)
+        G.add_edge(1, 2)
+        assert np.isclose(nx.number_of_spanning_trees(G, root=0), 2)
+
+    def test_nst_directed_selfloop(self):
+        G = nx.MultiDiGraph()
+        G = nx.cycle_graph(3, G)
+        G.add_edge(1, 1)
+        assert np.isclose(nx.number_of_spanning_trees(G, root=0), 1)
+
+    def test_nst_directed_weak_connected(self):
+        G = nx.MultiDiGraph()
+        G = nx.cycle_graph(3, G)
+        G.remove_edge(1, 2)
+        assert np.isclose(nx.number_of_spanning_trees(G, root=0), 0)
+
+    def test_nst_directed_weighted(self):
+        # from root=1:
+        # arborescence 1: 1->2, 1->3, weight=2*1
+        # arborescence 2: 1->2, 2->3, weight=2*3
+        G = nx.DiGraph()
+        G.add_edge(1, 2, weight=2)
+        G.add_edge(1, 3, weight=1)
+        G.add_edge(2, 3, weight=3)
+        Nst = nx.number_of_spanning_trees(G, root=1, weight="weight")
+        assert np.isclose(Nst, 8)
+        Nst = nx.number_of_spanning_trees(G, root=2, weight="weight")
+        assert np.isclose(Nst, 0)
+        Nst = nx.number_of_spanning_trees(G, root=3, weight="weight")
+        assert np.isclose(Nst, 0)
diff --git a/.venv/lib/python3.12/site-packages/networkx/algorithms/tree/tests/test_operations.py b/.venv/lib/python3.12/site-packages/networkx/algorithms/tree/tests/test_operations.py
new file mode 100644
index 00000000..284d94e2
--- /dev/null
+++ b/.venv/lib/python3.12/site-packages/networkx/algorithms/tree/tests/test_operations.py
@@ -0,0 +1,53 @@
+from itertools import chain
+
+import networkx as nx
+from networkx.utils import edges_equal, nodes_equal
+
+
+def _check_custom_label_attribute(input_trees, res_tree, label_attribute):
+    res_attr_dict = nx.get_node_attributes(res_tree, label_attribute)
+    res_attr_set = set(res_attr_dict.values())
+    input_label = (tree for tree, root in input_trees)
+    input_label_set = set(chain.from_iterable(input_label))
+    return res_attr_set == input_label_set
+
+
+def test_empty_sequence():
+    """Joining the empty sequence results in the tree with one node."""
+    T = nx.join_trees([])
+    assert len(T) == 1
+    assert T.number_of_edges() == 0
+
+
+def test_single():
+    """Joining just one tree yields a tree with one more node."""
+    T = nx.empty_graph(1)
+    trees = [(T, 0)]
+    actual_with_label = nx.join_trees(trees, label_attribute="custom_label")
+    expected = nx.path_graph(2)
+    assert nodes_equal(list(expected), list(actual_with_label))
+    assert edges_equal(list(expected.edges()), list(actual_with_label.edges()))
+
+
+def test_basic():
+    """Joining multiple subtrees at a root node."""
+    trees = [(nx.full_rary_tree(2, 2**2 - 1), 0) for i in range(2)]
+    expected = nx.full_rary_tree(2, 2**3 - 1)
+    actual = nx.join_trees(trees, label_attribute="old_labels")
+    assert nx.is_isomorphic(actual, expected)
+    assert _check_custom_label_attribute(trees, actual, "old_labels")
+
+    actual_without_label = nx.join_trees(trees)
+    assert nx.is_isomorphic(actual_without_label, expected)
+    # check that no labels were stored
+    assert all(not data for _, data in actual_without_label.nodes(data=True))
+
+
+def test_first_label():
+    """Test the functionality of the first_label argument."""
+    T1 = nx.path_graph(3)
+    T2 = nx.path_graph(2)
+    actual = nx.join_trees([(T1, 0), (T2, 0)], first_label=10)
+    expected_nodes = set(range(10, 16))
+    assert set(actual.nodes()) == expected_nodes
+    assert set(actual.neighbors(10)) == {11, 14}
diff --git a/.venv/lib/python3.12/site-packages/networkx/algorithms/tree/tests/test_recognition.py b/.venv/lib/python3.12/site-packages/networkx/algorithms/tree/tests/test_recognition.py
new file mode 100644
index 00000000..105f5a89
--- /dev/null
+++ b/.venv/lib/python3.12/site-packages/networkx/algorithms/tree/tests/test_recognition.py
@@ -0,0 +1,174 @@
+import pytest
+
+import networkx as nx
+
+
+class TestTreeRecognition:
+    graph = nx.Graph
+    multigraph = nx.MultiGraph
+
+    @classmethod
+    def setup_class(cls):
+        cls.T1 = cls.graph()
+
+        cls.T2 = cls.graph()
+        cls.T2.add_node(1)
+
+        cls.T3 = cls.graph()
+        cls.T3.add_nodes_from(range(5))
+        edges = [(i, i + 1) for i in range(4)]
+        cls.T3.add_edges_from(edges)
+
+        cls.T5 = cls.multigraph()
+        cls.T5.add_nodes_from(range(5))
+        edges = [(i, i + 1) for i in range(4)]
+        cls.T5.add_edges_from(edges)
+
+        cls.T6 = cls.graph()
+        cls.T6.add_nodes_from([6, 7])
+        cls.T6.add_edge(6, 7)
+
+        cls.F1 = nx.compose(cls.T6, cls.T3)
+
+        cls.N4 = cls.graph()
+        cls.N4.add_node(1)
+        cls.N4.add_edge(1, 1)
+
+        cls.N5 = cls.graph()
+        cls.N5.add_nodes_from(range(5))
+
+        cls.N6 = cls.graph()
+        cls.N6.add_nodes_from(range(3))
+        cls.N6.add_edges_from([(0, 1), (1, 2), (2, 0)])
+
+        cls.NF1 = nx.compose(cls.T6, cls.N6)
+
+    def test_null_tree(self):
+        with pytest.raises(nx.NetworkXPointlessConcept):
+            nx.is_tree(self.graph())
+
+    def test_null_tree2(self):
+        with pytest.raises(nx.NetworkXPointlessConcept):
+            nx.is_tree(self.multigraph())
+
+    def test_null_forest(self):
+        with pytest.raises(nx.NetworkXPointlessConcept):
+            nx.is_forest(self.graph())
+
+    def test_null_forest2(self):
+        with pytest.raises(nx.NetworkXPointlessConcept):
+            nx.is_forest(self.multigraph())
+
+    def test_is_tree(self):
+        assert nx.is_tree(self.T2)
+        assert nx.is_tree(self.T3)
+        assert nx.is_tree(self.T5)
+
+    def test_is_not_tree(self):
+        assert not nx.is_tree(self.N4)
+        assert not nx.is_tree(self.N5)
+        assert not nx.is_tree(self.N6)
+
+    def test_is_forest(self):
+        assert nx.is_forest(self.T2)
+        assert nx.is_forest(self.T3)
+        assert nx.is_forest(self.T5)
+        assert nx.is_forest(self.F1)
+        assert nx.is_forest(self.N5)
+
+    def test_is_not_forest(self):
+        assert not nx.is_forest(self.N4)
+        assert not nx.is_forest(self.N6)
+        assert not nx.is_forest(self.NF1)
+
+
+class TestDirectedTreeRecognition(TestTreeRecognition):
+    graph = nx.DiGraph
+    multigraph = nx.MultiDiGraph
+
+
+def test_disconnected_graph():
+    # https://github.com/networkx/networkx/issues/1144
+    G = nx.Graph()
+    G.add_edges_from([(0, 1), (1, 2), (2, 0), (3, 4)])
+    assert not nx.is_tree(G)
+
+    G = nx.DiGraph()
+    G.add_edges_from([(0, 1), (1, 2), (2, 0), (3, 4)])
+    assert not nx.is_tree(G)
+
+
+def test_dag_nontree():
+    G = nx.DiGraph()
+    G.add_edges_from([(0, 1), (0, 2), (1, 2)])
+    assert not nx.is_tree(G)
+    assert nx.is_directed_acyclic_graph(G)
+
+
+def test_multicycle():
+    G = nx.MultiDiGraph()
+    G.add_edges_from([(0, 1), (0, 1)])
+    assert not nx.is_tree(G)
+    assert nx.is_directed_acyclic_graph(G)
+
+
+def test_emptybranch():
+    G = nx.DiGraph()
+    G.add_nodes_from(range(10))
+    assert nx.is_branching(G)
+    assert not nx.is_arborescence(G)
+
+
+def test_is_branching_empty_graph_raises():
+    G = nx.DiGraph()
+    with pytest.raises(nx.NetworkXPointlessConcept, match="G has no nodes."):
+        nx.is_branching(G)
+
+
+def test_path():
+    G = nx.DiGraph()
+    nx.add_path(G, range(5))
+    assert nx.is_branching(G)
+    assert nx.is_arborescence(G)
+
+
+def test_notbranching1():
+    # Acyclic violation.
+    G = nx.MultiDiGraph()
+    G.add_nodes_from(range(10))
+    G.add_edges_from([(0, 1), (1, 0)])
+    assert not nx.is_branching(G)
+    assert not nx.is_arborescence(G)
+
+
+def test_notbranching2():
+    # In-degree violation.
+    G = nx.MultiDiGraph()
+    G.add_nodes_from(range(10))
+    G.add_edges_from([(0, 1), (0, 2), (3, 2)])
+    assert not nx.is_branching(G)
+    assert not nx.is_arborescence(G)
+
+
+def test_notarborescence1():
+    # Not an arborescence due to not spanning.
+    G = nx.MultiDiGraph()
+    G.add_nodes_from(range(10))
+    G.add_edges_from([(0, 1), (0, 2), (1, 3), (5, 6)])
+    assert nx.is_branching(G)
+    assert not nx.is_arborescence(G)
+
+
+def test_notarborescence2():
+    # Not an arborescence due to in-degree violation.
+    G = nx.MultiDiGraph()
+    nx.add_path(G, range(5))
+    G.add_edge(6, 4)
+    assert not nx.is_branching(G)
+    assert not nx.is_arborescence(G)
+
+
+def test_is_arborescense_empty_graph_raises():
+    G = nx.DiGraph()
+    with pytest.raises(nx.NetworkXPointlessConcept, match="G has no nodes."):
+        nx.is_arborescence(G)