code: def _select_jd_forests(n, q, cache_forests, seed):
      """Given `n` and `q`, returns a pair of positive integers $(j,d)$
      such that $j\\leq d$, with probability satisfying (F1) of [1]_.
  
      Parameters
      ----------
      n : int
          The number of nodes.
      q : int
          The maximum number of nodes for each tree of the forest.
      cache_forests : list of ints
          Cache for :func:`_num_rooted_forests`.
      seed : random_state
          See :ref:`Randomness<randomness>`.
  
      Returns
      -------
      (int, int)
          A pair of positive integers $(j,d)$
  
      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
      """
      p = seed.randint(
          0, _num_rooted_forests(n, q, cache_forests) * n - 1
      )
      cumsum = 0
      for d in range(q, 0, -1):
          for j in range(1, n // d + 1):
              cumsum += (
                  d
                  * _num_rooted_forests(n - j * d, q, cache_forests)
                  * _num_rooted_forests(d - 1, q, cache_forests)
              )
              if p < cumsum:
                  return (j, d)
  
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 13 for calling a global function with example usage: _select_jd_forests(n, q, cache_forests, seed) and returns: j,  d