Lengths of paths in rooted trees

TL;DR

This paper derives formulas for generating functions of paths in rooted trees, computes moments, and finds exact and asymptotic expectations for specific tree families, revealing surprisingly simple closed-form solutions.

Contribution

It introduces new formulas for path generating functions, computes moments, and provides exact and asymptotic expectations for particular rooted tree families.

Findings

Exact formulas for generating functions of paths.

Closed-form expressions for expectations in specific trees.

Asymptotic analysis of path expectations.

Abstract

We provide formulas for generating functions of many types of paths in various rooted tree structures. We compute the $k$th moment of the generating functions for various types of vertical paths. In two specific familes of trees we find exact closed formulas for expectations and their asymptotic values. Some of these closed formulas are surprisingly simple.