code: def _num_trees(n, cache_trees):
      """Returns the number of unlabeled trees with `n` nodes.
  
      See also https://oeis.org/A000055.
  
      Parameters
      ----------
      n : int
          The number of nodes.
      cache_trees : list of ints
          Cache for :func:`_num_rooted_trees`.
  
      Returns
      -------
      int
          The number of unlabeled trees with `n` nodes.
      """
      r = _num_rooted_trees(n, cache_trees) - sum(
          [
              _num_rooted_trees(j, cache_trees)
              * _num_rooted_trees(n - j, cache_trees)
              for j in range(1, n // 2 + 1)
          ]
      )
      if n % 2 == 0:
          r += comb(_num_rooted_trees(n // 2, cache_trees) + 1, 2)
      return r
  
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 16 for calling a global function with example usage: _num_trees(n, cache_trees) and returns: r