Comment by qazxcvbnm

For those who are not aware, knowing how to count the number of structures is (nearly) the same as knowing how to encode said structure as an integer (space-optimally) (the naive general algorithm would not have good time complexity); to convert to integer, simply define some ordering on said structures, enumerate until you find your structure, the index of the structure is the integer encoding; conversely, to decode, simply list all structures and pick the structure with the index of the integer.