code: 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
  
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 17 for calling a global function with example usage: _bicenter(n, cache, seed) and returns: t,  unknown