When solving this combinatorial problem, I learned a new thing called Catalan number. The problem is to find all isomorphic forms of binary trees using N nodes.
With simple divide and conquer, we can have the form:
F(0) = 1,
F(N) = sigma(0, N-1, F(i)*F(N-1-i))
Actually, F(N) is indeed the famous Catalan number. Catalan number has closed form representation:
C(N) = (2N, N)
However, it is non-trivial to get the closed form. The proof requires the knowledge of using generating function and non-integer binomial expansion to solve sequence sum. I found great resources of the proof.
Catalan number has other interesting applications