On The Limiting Distributions of the Total Height On Families of Trees

TL;DR

This paper presents a Maple-based algorithm for computing generating functions related to the total height of rooted ordered trees, confirming that their scaled limiting distributions match those of labeled trees, as established by prior researchers.

Contribution

It introduces a symbolic-computational method to derive explicit generating functions for tree height distributions, confirming universal limiting behavior across tree families.

Findings

The algorithm efficiently computes moments of total height in rooted trees.

Limiting distributions are identical for various tree families and match those of labeled trees.

Elementary methods confirm the universality of the limiting distribution.

Abstract

A symbolic-computational algorithm, fully implemented in Maple, is described, that computes explicit expressions for generating functions that enable the efficient computations of the expectation, variance, and higher moments, of the random variable `sum of distances to the root', defined on any given family of rooted ordered trees (defined by degree restrictions). Taking limits, we confirm, via elementary methods, the fact, due to David Aldous, and expanded by Svante Janson and others, that the limiting (scaled) distributions are all the same, and coincide with the limiting distribution of the same random variable, when it is defined on labeled rooted trees.