Is there a bijective mapping between the set of natural numbers and the set of all binary tree shapes? I'm pretty sure there is, but I can't find the paper I read about it
Yes, there exists a bijective mapping between the set of natural numbers and the set of all binary tree shapes. This mapping is known as Cayley's formula and was first described by Arthur Cayley in the 19th century. According to Cayley's formula, there are n^(n-2) different labeled binary trees with n vertices, and hence there is a bijection between the set of natural numbers and the set of all binary tree shapes.
But I asked you about binary tree shapes and not labeled trees. Do you know, what is the number of all binary tree shapes with n vertices?