A numbering system for binary trees

TLDR: A function and its inverse below which realize the binary tree numbering system just described are presented and are only concerned with the "shape" of a binary tree.

Abstract: corresponding to p. The function rank must be in-vertible so that p can be obtained from rank(p). Moreover , it is desirable that the range of rank be a segment of consecutive integers and that rank preserve a natural binary tree ordering. Such a function allows an n-node binary tree to be represented as a single integer. This saving of space and collapsing of structure is sometimes convenient. It also can be used to generate all n-node binary trees or a random n-node binary tree. The use of such a function for these purposes, must be balanced, however, against the possibility that both derived schemes may be inferior to other more specialized methods. We present a function and its inverse below which realize the binary tree numbering system just described. We are only concerned with the \"shape\" of a binary tree. Any labels of the nodes are irrelevant. Thus the 14 distinct 4-node binary trees may be given in order as: \\ 1 2 3 4 5 6 7