code: 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
  
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 14 for calling a global function with example usage: _random_unlabeled_rooted_forest(n, q, cache_trees, cache_forests, seed) and returns: t1,  t1_nodes,  r1, list,  int,  list