F - Huge Number of Trees

Hermione needs to solve a tough homework problem in Arithmancy class. Professor Vector was discussing trees for some time and now he gives the problem: Say, there are **n** + 1 nodes numbered 0, 1, 2, ..., **n**. Node 0 is always the root of the tree. How many different trees can be formed using the remaining **n** nodes so that all leaves have the same *depth* and the *degree* of each non-leaf no