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.
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