Leaf-to-leaf distances and their moments in finite and infinite m-ary tree graphs

TL;DR

This paper derives explicit formulas for leaf-to-leaf distances in m-ary trees, including average and higher moments, using recursive methods and special functions, with additional results for periodic and incomplete trees.

Contribution

It introduces a recursive approach to analytically compute leaf-to-leaf distances and their moments in finite and infinite m-ary trees, extending to periodic and incomplete cases.

Findings

Explicit formulas for total path sums and average distances

Expressions involving Hurwitz-Lerch transcendants

Rapid distance drop in incomplete binary trees for large r

Abstract

We study the leaf-to-leaf distances on full and complete m-ary graphs using a recursive approach. In our formulation, leaves are ordered along a line. We find explicit analytical formulae for the sum of all paths for arbitrary leaf-to-leaf distance r as well as the average path lengths and the moments thereof. We show that the resulting explicit expressions can be recast in terms of Hurwitz-Lerch transcendants. Results for periodic trees are also given. For incomplete random binary trees, we provide first results by numerical techniques; we find a rapid drop of leaf-to-leaf distances for large r.