code: 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
  
imports: import warnings
  from collections import Counter, defaultdict
  from math import comb, factorial
  import networkx
  from networkx.utils import py_random_state
label_desc: Use image node_id 10 for calling a global function with example usage: _random_unlabeled_rooted_tree(n, cache_trees, seed) and returns: t1,  t1_nodes, edges,  n_nodes, edges,  n_nodes