code: 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
  
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 18 for calling a global function with example usage: _random_unlabeled_tree(n, cache_trees, cache_forests, seed) and returns: _bicenter, f,  unknown