Tree Calculus for Bivariable Difference Equations

TL;DR

This paper develops a Tree Calculus to solve a system of partial difference equations related to binary trees, revealing the symmetry and explicit generating function of two key tree statistics.

Contribution

It introduces a novel Tree Calculus approach for analyzing binary trees with respect to two parameters and derives their joint distribution explicitly.

Findings

Joint distribution of the two parameters is symmetric.

Explicit three-variable generating function is obtained.

Method provides a systematic way to study related tree statistics.

Abstract

Following Poupard's study of strictly ordered binary trees with respect to two parameters, namely, "end of minimal chain" and "parent of maximum leaf" a true Tree Calculus is being developed to solve a partial difference equation system and then make a joint study of those two statistics. Their joint distribution is shown to be symmetric and to be expressed in the form of an explicit three-variable generating function.