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+"""Functions for generating trees.
+
+The functions sampling trees at random in this module come
+in two variants: labeled and unlabeled. The labeled variants
+sample from every possible tree with the given number of nodes
+uniformly at random. The unlabeled variants sample from every
+possible *isomorphism class* of trees with the given number
+of nodes uniformly at random.
+
+To understand the difference, consider the following example.
+There are two isomorphism classes of trees with four nodes.
+One is that of the path graph, the other is that of the
+star graph. The unlabeled variant will return a line graph or
+a star graph with probability 1/2.
+
+The labeled variant will return the line graph
+with probability 3/4 and the star graph with probability 1/4,
+because there are more labeled variants of the line graph
+than of the star graph. More precisely, the line graph has
+an automorphism group of order 2, whereas the star graph has
+an automorphism group of order 6, so the line graph has three
+times as many labeled variants as the star graph, and thus
+three more chances to be drawn.
+
+Additionally, some functions in this module can sample rooted
+trees and forests uniformly at random. A rooted tree is a tree
+with a designated root node. A rooted forest is a disjoint union
+of rooted trees.
+"""
+
+import warnings
+from collections import Counter, defaultdict
+from math import comb, factorial
+
+import networkx as nx
+from networkx.utils import py_random_state
+
+__all__ = [
+    "prefix_tree",
+    "prefix_tree_recursive",
+    "random_labeled_tree",
+    "random_labeled_rooted_tree",
+    "random_labeled_rooted_forest",
+    "random_unlabeled_tree",
+    "random_unlabeled_rooted_tree",
+    "random_unlabeled_rooted_forest",
+]
+
+
+@nx._dispatchable(graphs=None, returns_graph=True)
+def prefix_tree(paths):
+    """Creates a directed prefix tree from a list of paths.
+
+    Usually the paths are described as strings or lists of integers.
+
+    A "prefix tree" represents the prefix structure of the strings.
+    Each node represents a prefix of some string. The root represents
+    the empty prefix with children for the single letter prefixes which
+    in turn have children for each double letter prefix starting with
+    the single letter corresponding to the parent node, and so on.
+
+    More generally the prefixes do not need to be strings. A prefix refers
+    to the start of a sequence. The root has children for each one element
+    prefix and they have children for each two element prefix that starts
+    with the one element sequence of the parent, and so on.
+
+    Note that this implementation uses integer nodes with an attribute.
+    Each node has an attribute "source" whose value is the original element
+    of the path to which this node corresponds. For example, suppose `paths`
+    consists of one path: "can". Then the nodes `[1, 2, 3]` which represent
+    this path have "source" values "c", "a" and "n".
+
+    All the descendants of a node have a common prefix in the sequence/path
+    associated with that node. From the returned tree, the prefix for each
+    node can be constructed by traversing the tree up to the root and
+    accumulating the "source" values along the way.
+
+    The root node is always `0` and has "source" attribute `None`.
+    The root is the only node with in-degree zero.
+    The nil node is always `-1` and has "source" attribute `"NIL"`.
+    The nil node is the only node with out-degree zero.
+
+
+    Parameters
+    ----------
+    paths: iterable of paths
+        An iterable of paths which are themselves sequences.
+        Matching prefixes among these sequences are identified with
+        nodes of the prefix tree. One leaf of the tree is associated
+        with each path. (Identical paths are associated with the same
+        leaf of the tree.)
+
+
+    Returns
+    -------
+    tree: DiGraph
+        A directed graph representing an arborescence consisting of the
+        prefix tree generated by `paths`. Nodes are directed "downward",
+        from parent to child. A special "synthetic" root node is added
+        to be the parent of the first node in each path. A special
+        "synthetic" leaf node, the "nil" node `-1`, is added to be the child
+        of all nodes representing the last element in a path. (The
+        addition of this nil node technically makes this not an
+        arborescence but a directed acyclic graph; removing the nil node
+        makes it an arborescence.)
+
+
+    Notes
+    -----
+    The prefix tree is also known as a *trie*.
+
+
+    Examples
+    --------
+    Create a prefix tree from a list of strings with common prefixes::
+
+        >>> paths = ["ab", "abs", "ad"]
+        >>> T = nx.prefix_tree(paths)
+        >>> list(T.edges)
+        [(0, 1), (1, 2), (1, 4), (2, -1), (2, 3), (3, -1), (4, -1)]
+
+    The leaf nodes can be obtained as predecessors of the nil node::
+
+        >>> root, NIL = 0, -1
+        >>> list(T.predecessors(NIL))
+        [2, 3, 4]
+
+    To recover the original paths that generated the prefix tree,
+    traverse up the tree from the node `-1` to the node `0`::
+
+        >>> recovered = []
+        >>> for v in T.predecessors(NIL):
+        ...     prefix = ""
+        ...     while v != root:
+        ...         prefix = str(T.nodes[v]["source"]) + prefix
+        ...         v = next(T.predecessors(v))  # only one predecessor
+        ...     recovered.append(prefix)
+        >>> sorted(recovered)
+        ['ab', 'abs', 'ad']
+    """
+
+    def get_children(parent, paths):
+        children = defaultdict(list)
+        # Populate dictionary with key(s) as the child/children of the root and
+        # value(s) as the remaining paths of the corresponding child/children
+        for path in paths:
+            # If path is empty, we add an edge to the NIL node.
+            if not path:
+                tree.add_edge(parent, NIL)
+                continue
+            child, *rest = path
+            # `child` may exist as the head of more than one path in `paths`.
+            children[child].append(rest)
+        return children
+
+    # Initialize the prefix tree with a root node and a nil node.
+    tree = nx.DiGraph()
+    root = 0
+    tree.add_node(root, source=None)
+    NIL = -1
+    tree.add_node(NIL, source="NIL")
+    children = get_children(root, paths)
+    stack = [(root, iter(children.items()))]
+    while stack:
+        parent, remaining_children = stack[-1]
+        try:
+            child, remaining_paths = next(remaining_children)
+        # Pop item off stack if there are no remaining children
+        except StopIteration:
+            stack.pop()
+            continue
+        # We relabel each child with an unused name.
+        new_name = len(tree) - 1
+        # The "source" node attribute stores the original node name.
+        tree.add_node(new_name, source=child)
+        tree.add_edge(parent, new_name)
+        children = get_children(new_name, remaining_paths)
+        stack.append((new_name, iter(children.items())))
+
+    return tree
+
+
+@nx._dispatchable(graphs=None, returns_graph=True)
+def prefix_tree_recursive(paths):
+    """Recursively creates a directed prefix tree from a list of paths.
+
+    The original recursive version of prefix_tree for comparison. It is
+    the same algorithm but the recursion is unrolled onto a stack.
+
+    Usually the paths are described as strings or lists of integers.
+
+    A "prefix tree" represents the prefix structure of the strings.
+    Each node represents a prefix of some string. The root represents
+    the empty prefix with children for the single letter prefixes which
+    in turn have children for each double letter prefix starting with
+    the single letter corresponding to the parent node, and so on.
+
+    More generally the prefixes do not need to be strings. A prefix refers
+    to the start of a sequence. The root has children for each one element
+    prefix and they have children for each two element prefix that starts
+    with the one element sequence of the parent, and so on.
+
+    Note that this implementation uses integer nodes with an attribute.
+    Each node has an attribute "source" whose value is the original element
+    of the path to which this node corresponds. For example, suppose `paths`
+    consists of one path: "can". Then the nodes `[1, 2, 3]` which represent
+    this path have "source" values "c", "a" and "n".
+
+    All the descendants of a node have a common prefix in the sequence/path
+    associated with that node. From the returned tree, ehe prefix for each
+    node can be constructed by traversing the tree up to the root and
+    accumulating the "source" values along the way.
+
+    The root node is always `0` and has "source" attribute `None`.
+    The root is the only node with in-degree zero.
+    The nil node is always `-1` and has "source" attribute `"NIL"`.
+    The nil node is the only node with out-degree zero.
+
+
+    Parameters
+    ----------
+    paths: iterable of paths
+        An iterable of paths which are themselves sequences.
+        Matching prefixes among these sequences are identified with
+        nodes of the prefix tree. One leaf of the tree is associated
+        with each path. (Identical paths are associated with the same
+        leaf of the tree.)
+
+
+    Returns
+    -------
+    tree: DiGraph
+        A directed graph representing an arborescence consisting of the
+        prefix tree generated by `paths`. Nodes are directed "downward",
+        from parent to child. A special "synthetic" root node is added
+        to be the parent of the first node in each path. A special
+        "synthetic" leaf node, the "nil" node `-1`, is added to be the child
+        of all nodes representing the last element in a path. (The
+        addition of this nil node technically makes this not an
+        arborescence but a directed acyclic graph; removing the nil node
+        makes it an arborescence.)
+
+
+    Notes
+    -----
+    The prefix tree is also known as a *trie*.
+
+
+    Examples
+    --------
+    Create a prefix tree from a list of strings with common prefixes::
+
+        >>> paths = ["ab", "abs", "ad"]
+        >>> T = nx.prefix_tree(paths)
+        >>> list(T.edges)
+        [(0, 1), (1, 2), (1, 4), (2, -1), (2, 3), (3, -1), (4, -1)]
+
+    The leaf nodes can be obtained as predecessors of the nil node.
+
+        >>> root, NIL = 0, -1
+        >>> list(T.predecessors(NIL))
+        [2, 3, 4]
+
+    To recover the original paths that generated the prefix tree,
+    traverse up the tree from the node `-1` to the node `0`::
+
+        >>> recovered = []
+        >>> for v in T.predecessors(NIL):
+        ...     prefix = ""
+        ...     while v != root:
+        ...         prefix = str(T.nodes[v]["source"]) + prefix
+        ...         v = next(T.predecessors(v))  # only one predecessor
+        ...     recovered.append(prefix)
+        >>> sorted(recovered)
+        ['ab', 'abs', 'ad']
+    """
+
+    def _helper(paths, root, tree):
+        """Recursively create a trie from the given list of paths.
+
+        `paths` is a list of paths, each of which is itself a list of
+        nodes, relative to the given `root` (but not including it). This
+        list of paths will be interpreted as a tree-like structure, in
+        which two paths that share a prefix represent two branches of
+        the tree with the same initial segment.
+
+        `root` is the parent of the node at index 0 in each path.
+
+        `tree` is the "accumulator", the :class:`networkx.DiGraph`
+        representing the branching to which the new nodes and edges will
+        be added.
+
+        """
+        # For each path, remove the first node and make it a child of root.
+        # Any remaining paths then get processed recursively.
+        children = defaultdict(list)
+        for path in paths:
+            # If path is empty, we add an edge to the NIL node.
+            if not path:
+                tree.add_edge(root, NIL)
+                continue
+            child, *rest = path
+            # `child` may exist as the head of more than one path in `paths`.
+            children[child].append(rest)
+        # Add a node for each child, connect root, recurse to remaining paths
+        for child, remaining_paths in children.items():
+            # We relabel each child with an unused name.
+            new_name = len(tree) - 1
+            # The "source" node attribute stores the original node name.
+            tree.add_node(new_name, source=child)
+            tree.add_edge(root, new_name)
+            _helper(remaining_paths, new_name, tree)
+
+    # Initialize the prefix tree with a root node and a nil node.
+    tree = nx.DiGraph()
+    root = 0
+    tree.add_node(root, source=None)
+    NIL = -1
+    tree.add_node(NIL, source="NIL")
+    # Populate the tree.
+    _helper(paths, root, tree)
+    return tree
+
+
+@py_random_state("seed")
+@nx._dispatchable(graphs=None, returns_graph=True)
+def random_labeled_tree(n, *, seed=None):
+    """Returns a labeled tree on `n` nodes chosen uniformly at random.
+
+    Generating uniformly distributed random Prüfer sequences and
+    converting them into the corresponding trees is a straightforward
+    method of generating uniformly distributed random labeled trees.
+    This function implements this method.
+
+    Parameters
+    ----------
+    n : int
+        The number of nodes, greater than zero.
+    seed : random_state
+        Indicator of random number generation state.
+        See :ref:`Randomness<randomness>`
+
+    Returns
+    -------
+     :class:`networkx.Graph`
+        A `networkx.Graph` with nodes in the set {0, …, *n* - 1}.
+
+    Raises
+    ------
+    NetworkXPointlessConcept
+        If `n` is zero (because the null graph is not a tree).
+
+    Examples
+    --------
+    >>> G = nx.random_labeled_tree(5, seed=42)
+    >>> nx.is_tree(G)
+    True
+    >>> G.edges
+    EdgeView([(0, 1), (0, 3), (0, 2), (2, 4)])
+
+    A tree with *arbitrarily directed* edges can be created by assigning
+    generated edges to a ``DiGraph``:
+
+    >>> DG = nx.DiGraph()
+    >>> DG.add_edges_from(G.edges)
+    >>> nx.is_tree(DG)
+    True
+    >>> DG.edges
+    OutEdgeView([(0, 1), (0, 3), (0, 2), (2, 4)])
+    """
+    # Cannot create a Prüfer sequence unless `n` is at least two.
+    if n == 0:
+        raise nx.NetworkXPointlessConcept("the null graph is not a tree")
+    if n == 1:
+        return nx.empty_graph(1)
+    return nx.from_prufer_sequence([seed.choice(range(n)) for i in range(n - 2)])
+
+
+@py_random_state("seed")
+@nx._dispatchable(graphs=None, returns_graph=True)
+def random_labeled_rooted_tree(n, *, seed=None):
+    """Returns a labeled rooted tree with `n` nodes.
+
+    The returned tree is chosen uniformly at random from all labeled rooted trees.
+
+    Parameters
+    ----------
+    n : int
+        The number of nodes
+    seed : integer, random_state, or None (default)
+        Indicator of random number generation state.
+        See :ref:`Randomness<randomness>`.
+
+    Returns
+    -------
+    :class:`networkx.Graph`
+        A `networkx.Graph` with integer nodes 0 <= node <= `n` - 1.
+        The root of the tree is selected uniformly from the nodes.
+        The "root" graph attribute identifies the root of the tree.
+
+    Notes
+    -----
+    This function returns the result of :func:`random_labeled_tree`
+    with a randomly selected root.
+
+    Raises
+    ------
+    NetworkXPointlessConcept
+        If `n` is zero (because the null graph is not a tree).
+    """
+    t = random_labeled_tree(n, seed=seed)
+    t.graph["root"] = seed.randint(0, n - 1)
+    return t
+
+
+@py_random_state("seed")
+@nx._dispatchable(graphs=None, returns_graph=True)
+def random_labeled_rooted_forest(n, *, seed=None):
+    """Returns a labeled rooted forest with `n` nodes.
+
+    The returned forest is chosen uniformly at random using a
+    generalization of Prüfer sequences [1]_ in the form described in [2]_.
+
+    Parameters
+    ----------
+    n : int
+        The number of nodes.
+    seed : random_state
+       See :ref:`Randomness<randomness>`.
+
+    Returns
+    -------
+    :class:`networkx.Graph`
+        A `networkx.Graph` with integer nodes 0 <= node <= `n` - 1.
+        The "roots" graph attribute is a set of integers containing the roots.
+
+    References
+    ----------
+    .. [1] Knuth, Donald E. "Another Enumeration of Trees."
+        Canadian Journal of Mathematics, 20 (1968): 1077-1086.
+        https://doi.org/10.4153/CJM-1968-104-8
+    .. [2] Rubey, Martin. "Counting Spanning Trees". Diplomarbeit
+        zur Erlangung des akademischen Grades Magister der
+        Naturwissenschaften an der Formal- und Naturwissenschaftlichen
+        Fakultät der Universität Wien. Wien, May 2000.
+    """
+
+    # Select the number of roots by iterating over the cumulative count of trees
+    # with at most k roots
+    def _select_k(n, seed):
+        r = seed.randint(0, (n + 1) ** (n - 1) - 1)
+        cum_sum = 0
+        for k in range(1, n):
+            cum_sum += (factorial(n - 1) * n ** (n - k)) // (
+                factorial(k - 1) * factorial(n - k)
+            )
+            if r < cum_sum:
+                return k
+
+        return n
+
+    F = nx.empty_graph(n)
+    if n == 0:
+        F.graph["roots"] = {}
+        return F
+    # Select the number of roots k
+    k = _select_k(n, seed)
+    if k == n:
+        F.graph["roots"] = set(range(n))
+        return F  # Nothing to do
+    # Select the roots
+    roots = seed.sample(range(n), k)
+    # Nonroots
+    p = set(range(n)).difference(roots)
+    # Coding sequence
+    N = [seed.randint(0, n - 1) for i in range(n - k - 1)]
+    # Multiset of elements in N also in p
+    degree = Counter([x for x in N if x in p])
+    # Iterator over the elements of p with degree zero
+    iterator = iter(x for x in p if degree[x] == 0)
+    u = last = next(iterator)
+    # This loop is identical to that for Prüfer sequences,
+    # except that we can draw nodes only from p
+    for v in N:
+        F.add_edge(u, v)
+        degree[v] -= 1
+        if v < last and degree[v] == 0:
+            u = v
+        else:
+            last = u = next(iterator)
+
+    F.add_edge(u, roots[0])
+    F.graph["roots"] = set(roots)
+    return F
+
+
+# The following functions support generation of unlabeled trees and forests.
+
+
+def _to_nx(edges, n_nodes, root=None, roots=None):
+    """
+    Converts the (edges, n_nodes) input to a :class:`networkx.Graph`.
+    The (edges, n_nodes) input is a list of even length, where each pair
+    of consecutive integers represents an edge, and an integer `n_nodes`.
+    Integers in the list are elements of `range(n_nodes)`.
+
+    Parameters
+    ----------
+    edges : list of ints
+        The flattened list of edges of the graph.
+    n_nodes : int
+        The number of nodes of the graph.
+    root: int (default=None)
+        If not None, the "root" attribute of the graph will be set to this value.
+    roots: collection of ints (default=None)
+        If not None, he "roots" attribute of the graph will be set to this value.
+
+    Returns
+    -------
+    :class:`networkx.Graph`
+        The graph with `n_nodes` nodes and edges given by `edges`.
+    """
+    G = nx.empty_graph(n_nodes)
+    G.add_edges_from(edges)
+    if root is not None:
+        G.graph["root"] = root
+    if roots is not None:
+        G.graph["roots"] = roots
+    return G
+
+
+def _num_rooted_trees(n, cache_trees):
+    """Returns the number of unlabeled rooted trees with `n` nodes.
+
+    See also https://oeis.org/A000081.
+
+    Parameters
+    ----------
+    n : int
+        The number of nodes
+    cache_trees : list of ints
+        The $i$-th element is the number of unlabeled rooted trees with $i$ nodes,
+        which is used as a cache (and is extended to length $n+1$ if needed)
+
+    Returns
+    -------
+    int
+        The number of unlabeled rooted trees with `n` nodes.
+    """
+    for n_i in range(len(cache_trees), n + 1):
+        cache_trees.append(
+            sum(
+                [
+                    d * cache_trees[n_i - j * d] * cache_trees[d]
+                    for d in range(1, n_i)
+                    for j in range(1, (n_i - 1) // d + 1)
+                ]
+            )
+            // (n_i - 1)
+        )
+    return cache_trees[n]
+
+
+def _select_jd_trees(n, cache_trees, seed):
+    """Returns a pair $(j,d)$ with a specific probability
+
+    Given $n$, returns a pair of positive integers $(j,d)$ with the probability
+    specified in formula (5) of Chapter 29 of [1]_.
+
+    Parameters
+    ----------
+    n : int
+        The number of nodes
+    cache_trees : list of ints
+        Cache for :func:`_num_rooted_trees`.
+    seed : random_state
+       See :ref:`Randomness<randomness>`.
+
+    Returns
+    -------
+    (int, int)
+        A pair of positive integers $(j,d)$ satisfying formula (5) of
+        Chapter 29 of [1]_.
+
+    References
+    ----------
+    .. [1] Nijenhuis, Albert, and Wilf, Herbert S.
+        "Combinatorial algorithms: for computers and calculators."
+        Academic Press, 1978.
+        https://doi.org/10.1016/C2013-0-11243-3
+    """
+    p = seed.randint(0, _num_rooted_trees(n, cache_trees) * (n - 1) - 1)
+    cumsum = 0
+    for d in range(n - 1, 0, -1):
+        for j in range(1, (n - 1) // d + 1):
+            cumsum += (
+                d
+                * _num_rooted_trees(n - j * d, cache_trees)
+                * _num_rooted_trees(d, cache_trees)
+            )
+            if p < cumsum:
+                return (j, d)
+
+
+def _random_unlabeled_rooted_tree(n, cache_trees, seed):
+    """Returns an unlabeled rooted tree with `n` nodes.
+
+    Returns an unlabeled rooted tree with `n` nodes chosen uniformly
+    at random using the "RANRUT" algorithm from [1]_.
+    The tree is returned in the form: (list_of_edges, number_of_nodes)
+
+    Parameters
+    ----------
+    n : int
+        The number of nodes, greater than zero.
+    cache_trees : list ints
+        Cache for :func:`_num_rooted_trees`.
+    seed : random_state
+        See :ref:`Randomness<randomness>`.
+
+    Returns
+    -------
+    (list_of_edges, number_of_nodes) : list, int
+        A random unlabeled rooted tree with `n` nodes as a 2-tuple
+        ``(list_of_edges, number_of_nodes)``.
+        The root is node 0.
+
+    References
+    ----------
+    .. [1] Nijenhuis, Albert, and Wilf, Herbert S.
+        "Combinatorial algorithms: for computers and calculators."
+        Academic Press, 1978.
+        https://doi.org/10.1016/C2013-0-11243-3
+    """
+    if n == 1:
+        edges, n_nodes = [], 1
+        return edges, n_nodes
+    if n == 2:
+        edges, n_nodes = [(0, 1)], 2
+        return edges, n_nodes
+
+    j, d = _select_jd_trees(n, cache_trees, seed)
+    t1, t1_nodes = _random_unlabeled_rooted_tree(n - j * d, cache_trees, seed)
+    t2, t2_nodes = _random_unlabeled_rooted_tree(d, cache_trees, seed)
+    t12 = [(0, t2_nodes * i + t1_nodes) for i in range(j)]
+    t1.extend(t12)
+    for _ in range(j):
+        t1.extend((n1 + t1_nodes, n2 + t1_nodes) for n1, n2 in t2)
+        t1_nodes += t2_nodes
+
+    return t1, t1_nodes
+
+
+@py_random_state("seed")
+@nx._dispatchable(graphs=None, returns_graph=True)
+def random_unlabeled_rooted_tree(n, *, number_of_trees=None, seed=None):
+    """Returns a number of unlabeled rooted trees uniformly at random
+
+    Returns one or more (depending on `number_of_trees`)
+    unlabeled rooted trees with `n` nodes drawn uniformly
+    at random.
+
+    Parameters
+    ----------
+    n : int
+        The number of nodes
+    number_of_trees : int or None (default)
+        If not None, this number of trees is generated and returned.
+    seed : integer, random_state, or None (default)
+        Indicator of random number generation state.
+        See :ref:`Randomness<randomness>`.
+
+    Returns
+    -------
+    :class:`networkx.Graph` or list of :class:`networkx.Graph`
+        A single `networkx.Graph` (or a list thereof, if `number_of_trees`
+        is specified) with nodes in the set {0, …, *n* - 1}.
+        The "root" graph attribute identifies the root of the tree.
+
+    Notes
+    -----
+    The trees are generated using the "RANRUT" algorithm from [1]_.
+    The algorithm needs to compute some counting functions
+    that are relatively expensive: in case several trees are needed,
+    it is advisable to use the `number_of_trees` optional argument
+    to reuse the counting functions.
+
+    Raises
+    ------
+    NetworkXPointlessConcept
+        If `n` is zero (because the null graph is not a tree).
+
+    References
+    ----------
+    .. [1] Nijenhuis, Albert, and Wilf, Herbert S.
+        "Combinatorial algorithms: for computers and calculators."
+        Academic Press, 1978.
+        https://doi.org/10.1016/C2013-0-11243-3
+    """
+    if n == 0:
+        raise nx.NetworkXPointlessConcept("the null graph is not a tree")
+    cache_trees = [0, 1]  # initial cache of number of rooted trees
+    if number_of_trees is None:
+        return _to_nx(*_random_unlabeled_rooted_tree(n, cache_trees, seed), root=0)
+    return [
+        _to_nx(*_random_unlabeled_rooted_tree(n, cache_trees, seed), root=0)
+        for i in range(number_of_trees)
+    ]
+
+
+def _num_rooted_forests(n, q, cache_forests):
+    """Returns the number of unlabeled rooted forests with `n` nodes, and with
+    no more than `q` nodes per tree. A recursive formula for this is (2) in
+    [1]_. This function is implemented using dynamic programming instead of
+    recursion.
+
+    Parameters
+    ----------
+    n : int
+        The number of nodes.
+    q : int
+        The maximum number of nodes for each tree of the forest.
+    cache_forests : list of ints
+        The $i$-th element is the number of unlabeled rooted forests with
+        $i$ nodes, and with no more than `q` nodes per tree; this is used
+        as a cache (and is extended to length `n` + 1 if needed).
+
+    Returns
+    -------
+    int
+        The number of unlabeled rooted forests with `n` nodes with no more than
+        `q` nodes per tree.
+
+    References
+    ----------
+    .. [1] Wilf, Herbert S. "The uniform selection of free trees."
+        Journal of Algorithms 2.2 (1981): 204-207.
+        https://doi.org/10.1016/0196-6774(81)90021-3
+    """
+    for n_i in range(len(cache_forests), n + 1):
+        q_i = min(n_i, q)
+        cache_forests.append(
+            sum(
+                [
+                    d * cache_forests[n_i - j * d] * cache_forests[d - 1]
+                    for d in range(1, q_i + 1)
+                    for j in range(1, n_i // d + 1)
+                ]
+            )
+            // n_i
+        )
+
+    return cache_forests[n]
+
+
+def _select_jd_forests(n, q, cache_forests, seed):
+    """Given `n` and `q`, returns a pair of positive integers $(j,d)$
+    such that $j\\leq d$, with probability satisfying (F1) of [1]_.
+
+    Parameters
+    ----------
+    n : int
+        The number of nodes.
+    q : int
+        The maximum number of nodes for each tree of the forest.
+    cache_forests : list of ints
+        Cache for :func:`_num_rooted_forests`.
+    seed : random_state
+        See :ref:`Randomness<randomness>`.
+
+    Returns
+    -------
+    (int, int)
+        A pair of positive integers $(j,d)$
+
+    References
+    ----------
+    .. [1] Wilf, Herbert S. "The uniform selection of free trees."
+        Journal of Algorithms 2.2 (1981): 204-207.
+        https://doi.org/10.1016/0196-6774(81)90021-3
+    """
+    p = seed.randint(0, _num_rooted_forests(n, q, cache_forests) * n - 1)
+    cumsum = 0
+    for d in range(q, 0, -1):
+        for j in range(1, n // d + 1):
+            cumsum += (
+                d
+                * _num_rooted_forests(n - j * d, q, cache_forests)
+                * _num_rooted_forests(d - 1, q, cache_forests)
+            )
+            if p < cumsum:
+                return (j, d)
+
+
+def _random_unlabeled_rooted_forest(n, q, cache_trees, cache_forests, seed):
+    """Returns an unlabeled rooted forest with `n` nodes, and with no more
+    than `q` nodes per tree, drawn uniformly at random. It is an implementation
+    of the algorithm "Forest" of [1]_.
+
+    Parameters
+    ----------
+    n : int
+        The number of nodes.
+    q : int
+        The maximum number of nodes per tree.
+    cache_trees :
+        Cache for :func:`_num_rooted_trees`.
+    cache_forests :
+        Cache for :func:`_num_rooted_forests`.
+    seed : random_state
+       See :ref:`Randomness<randomness>`.
+
+    Returns
+    -------
+    (edges, n, r) : (list, int, list)
+        The forest (edges, n) and a list r of root nodes.
+
+    References
+    ----------
+    .. [1] Wilf, Herbert S. "The uniform selection of free trees."
+        Journal of Algorithms 2.2 (1981): 204-207.
+        https://doi.org/10.1016/0196-6774(81)90021-3
+    """
+    if n == 0:
+        return ([], 0, [])
+
+    j, d = _select_jd_forests(n, q, cache_forests, seed)
+    t1, t1_nodes, r1 = _random_unlabeled_rooted_forest(
+        n - j * d, q, cache_trees, cache_forests, seed
+    )
+    t2, t2_nodes = _random_unlabeled_rooted_tree(d, cache_trees, seed)
+    for _ in range(j):
+        r1.append(t1_nodes)
+        t1.extend((n1 + t1_nodes, n2 + t1_nodes) for n1, n2 in t2)
+        t1_nodes += t2_nodes
+    return t1, t1_nodes, r1
+
+
+@py_random_state("seed")
+@nx._dispatchable(graphs=None, returns_graph=True)
+def random_unlabeled_rooted_forest(n, *, q=None, number_of_forests=None, seed=None):
+    """Returns a forest or list of forests selected at random.
+
+    Returns one or more (depending on `number_of_forests`)
+    unlabeled rooted forests with `n` nodes, and with no more than
+    `q` nodes per tree, drawn uniformly at random.
+    The "roots" graph attribute identifies the roots of the forest.
+
+    Parameters
+    ----------
+    n : int
+        The number of nodes
+    q : int or None (default)
+        The maximum number of nodes per tree.
+    number_of_forests : int or None (default)
+        If not None, this number of forests is generated and returned.
+    seed : integer, random_state, or None (default)
+        Indicator of random number generation state.
+        See :ref:`Randomness<randomness>`.
+
+    Returns
+    -------
+    :class:`networkx.Graph` or list of :class:`networkx.Graph`
+        A single `networkx.Graph` (or a list thereof, if `number_of_forests`
+        is specified) with nodes in the set {0, …, *n* - 1}.
+        The "roots" graph attribute is a set containing the roots
+        of the trees in the forest.
+
+    Notes
+    -----
+    This function implements the algorithm "Forest" of [1]_.
+    The algorithm needs to compute some counting functions
+    that are relatively expensive: in case several trees are needed,
+    it is advisable to use the `number_of_forests` optional argument
+    to reuse the counting functions.
+
+    Raises
+    ------
+    ValueError
+        If `n` is non-zero but `q` is zero.
+
+    References
+    ----------
+    .. [1] Wilf, Herbert S. "The uniform selection of free trees."
+        Journal of Algorithms 2.2 (1981): 204-207.
+        https://doi.org/10.1016/0196-6774(81)90021-3
+    """
+    if q is None:
+        q = n
+    if q == 0 and n != 0:
+        raise ValueError("q must be a positive integer if n is positive.")
+
+    cache_trees = [0, 1]  # initial cache of number of rooted trees
+    cache_forests = [1]  # initial cache of number of rooted forests
+
+    if number_of_forests is None:
+        g, nodes, rs = _random_unlabeled_rooted_forest(
+            n, q, cache_trees, cache_forests, seed
+        )
+        return _to_nx(g, nodes, roots=set(rs))
+
+    res = []
+    for i in range(number_of_forests):
+        g, nodes, rs = _random_unlabeled_rooted_forest(
+            n, q, cache_trees, cache_forests, seed
+        )
+        res.append(_to_nx(g, nodes, roots=set(rs)))
+    return res
+
+
+def _num_trees(n, cache_trees):
+    """Returns the number of unlabeled trees with `n` nodes.
+
+    See also https://oeis.org/A000055.
+
+    Parameters
+    ----------
+    n : int
+        The number of nodes.
+    cache_trees : list of ints
+        Cache for :func:`_num_rooted_trees`.
+
+    Returns
+    -------
+    int
+        The number of unlabeled trees with `n` nodes.
+    """
+    r = _num_rooted_trees(n, cache_trees) - sum(
+        [
+            _num_rooted_trees(j, cache_trees) * _num_rooted_trees(n - j, cache_trees)
+            for j in range(1, n // 2 + 1)
+        ]
+    )
+    if n % 2 == 0:
+        r += comb(_num_rooted_trees(n // 2, cache_trees) + 1, 2)
+    return r
+
+
+def _bicenter(n, cache, seed):
+    """Returns a bi-centroidal tree on `n` nodes drawn uniformly at random.
+
+    This function implements the algorithm Bicenter of [1]_.
+
+    Parameters
+    ----------
+    n : int
+        The number of nodes (must be even).
+    cache : list of ints.
+        Cache for :func:`_num_rooted_trees`.
+    seed : random_state
+        See :ref:`Randomness<randomness>`
+
+    Returns
+    -------
+    (edges, n)
+        The tree as a list of edges and number of nodes.
+
+    References
+    ----------
+    .. [1] Wilf, Herbert S. "The uniform selection of free trees."
+        Journal of Algorithms 2.2 (1981): 204-207.
+        https://doi.org/10.1016/0196-6774(81)90021-3
+    """
+    t, t_nodes = _random_unlabeled_rooted_tree(n // 2, cache, seed)
+    if seed.randint(0, _num_rooted_trees(n // 2, cache)) == 0:
+        t2, t2_nodes = t, t_nodes
+    else:
+        t2, t2_nodes = _random_unlabeled_rooted_tree(n // 2, cache, seed)
+    t.extend([(n1 + (n // 2), n2 + (n // 2)) for n1, n2 in t2])
+    t.append((0, n // 2))
+    return t, t_nodes + t2_nodes
+
+
+def _random_unlabeled_tree(n, cache_trees, cache_forests, seed):
+    """Returns a tree on `n` nodes drawn uniformly at random.
+    It implements the Wilf's algorithm "Free" of [1]_.
+
+    Parameters
+    ----------
+    n : int
+        The number of nodes, greater than zero.
+    cache_trees : list of ints
+        Cache for :func:`_num_rooted_trees`.
+    cache_forests : list of ints
+        Cache for :func:`_num_rooted_forests`.
+    seed : random_state
+        Indicator of random number generation state.
+        See :ref:`Randomness<randomness>`
+
+    Returns
+    -------
+    (edges, n)
+        The tree as a list of edges and number of nodes.
+
+    References
+    ----------
+    .. [1] Wilf, Herbert S. "The uniform selection of free trees."
+        Journal of Algorithms 2.2 (1981): 204-207.
+        https://doi.org/10.1016/0196-6774(81)90021-3
+    """
+    if n % 2 == 1:
+        p = 0
+    else:
+        p = comb(_num_rooted_trees(n // 2, cache_trees) + 1, 2)
+    if seed.randint(0, _num_trees(n, cache_trees) - 1) < p:
+        return _bicenter(n, cache_trees, seed)
+    else:
+        f, n_f, r = _random_unlabeled_rooted_forest(
+            n - 1, (n - 1) // 2, cache_trees, cache_forests, seed
+        )
+        for i in r:
+            f.append((i, n_f))
+        return f, n_f + 1
+
+
+@py_random_state("seed")
+@nx._dispatchable(graphs=None, returns_graph=True)
+def random_unlabeled_tree(n, *, number_of_trees=None, seed=None):
+    """Returns a tree or list of trees chosen randomly.
+
+    Returns one or more (depending on `number_of_trees`)
+    unlabeled trees with `n` nodes drawn uniformly at random.
+
+    Parameters
+    ----------
+    n : int
+        The number of nodes
+    number_of_trees : int or None (default)
+        If not None, this number of trees is generated and returned.
+    seed : integer, random_state, or None (default)
+        Indicator of random number generation state.
+        See :ref:`Randomness<randomness>`.
+
+    Returns
+    -------
+    :class:`networkx.Graph` or list of :class:`networkx.Graph`
+        A single `networkx.Graph` (or a list thereof, if
+        `number_of_trees` is specified) with nodes in the set {0, …, *n* - 1}.
+
+    Raises
+    ------
+    NetworkXPointlessConcept
+        If `n` is zero (because the null graph is not a tree).
+
+    Notes
+    -----
+    This function generates an unlabeled tree uniformly at random using
+    Wilf's algorithm "Free" of [1]_. The algorithm needs to
+    compute some counting functions that are relatively expensive:
+    in case several trees are needed, it is advisable to use the
+    `number_of_trees` optional argument to reuse the counting
+    functions.
+
+    References
+    ----------
+    .. [1] Wilf, Herbert S. "The uniform selection of free trees."
+        Journal of Algorithms 2.2 (1981): 204-207.
+        https://doi.org/10.1016/0196-6774(81)90021-3
+    """
+    if n == 0:
+        raise nx.NetworkXPointlessConcept("the null graph is not a tree")
+
+    cache_trees = [0, 1]  # initial cache of number of rooted trees
+    cache_forests = [1]  # initial cache of number of rooted forests
+    if number_of_trees is None:
+        return _to_nx(*_random_unlabeled_tree(n, cache_trees, cache_forests, seed))
+    else:
+        return [
+            _to_nx(*_random_unlabeled_tree(n, cache_trees, cache_forests, seed))
+            for i in range(number_of_trees)
+        ]