code: def _select_jd_trees(n, cache_trees, seed):
      """Returns a pair $(j,d)$ with a specific probability
  
      Given $n$, returns a pair of positive integers $(j,d)$ with the probability
      specified in formula (5) of Chapter 29 of [1]_.
  
      Parameters
      ----------
      n : int
          The number of nodes
      cache_trees : list of ints
          Cache for :func:`_num_rooted_trees`.
      seed : random_state
         See :ref:`Randomness<randomness>`.
  
      Returns
      -------
      (int, int)
          A pair of positive integers $(j,d)$ satisfying formula (5) of
          Chapter 29 of [1]_.
  
      References
      ----------
      .. [1] Nijenhuis, Albert, and Wilf, Herbert S.
          "Combinatorial algorithms: for computers and calculators."
          Academic Press, 1978.
          https://doi.org/10.1016/C2013-0-11243-3
      """
      p = seed.randint(
          0, _num_rooted_trees(n, cache_trees) * (n - 1) - 1
      )
      cumsum = 0
      for d in range(n - 1, 0, -1):
          for j in range(1, (n - 1) // d + 1):
              cumsum += (
                  d
                  * _num_rooted_trees(n - j * d, cache_trees)
                  * _num_rooted_trees(d, cache_trees)
              )
              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 9 for calling a global function with example usage: _select_jd_trees(n, cache_trees, seed) and returns: j,  d