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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
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+"""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)