Dealing with balanced and unbalanced binary trees.
height = 0, possible trees = 1 (nothing)
height = 1, possible trees = 1 (leaf)
height = 2, possible trees = 3
I'm looking at the Catalan function but it has not lead me anywhere beneficial, mostly because I think it might be counting trees less than height h. For example, if height it 2, it will count height 1 too (and maybe height 0) I believe.
You appear to be looking for integer sequence A001699, "Number of binary trees of height n". One possible algorithm to generate them being:
a(n+1) = 2*a(n)*(a(0)+...+a(n-1))+a(n)^2
Unfortunately, there doesn't seem to be a closed-form version. Each formula is itself recursive, or uses A003095, which is also recursive.