The distribution of the number of automorphisms of random trees

TL;DR

This paper investigates the distribution of automorphism group sizes in random trees, showing they asymptotically follow a log-normal distribution and providing formulas for their mean and variance.

Contribution

It establishes the asymptotic log-normal distribution of automorphism group sizes for two types of random trees and extends results to some unrooted trees.

Findings

Automorphism group sizes follow a log-normal distribution asymptotically.

Derived asymptotic formulas for mean and variance of the logarithm of automorphism sizes.

Extended results to certain classes of unrooted trees.

Abstract

We study the size of the automorphism group of two different types of random trees: Galton--Watson trees and rooted P\'olya trees. In both cases, we prove that it asymptotically follows a log-normal distribution and provide asymptotic formulas for mean and variance of the logarithm of the size of the automorphism group. While the proof for Galton--Watson trees mainly relies on probabilistic arguments and a general result on additive tree functionals, generating functions are used in the case of rooted P\'olya trees. We also show how to extend the results to some classes of unrooted trees.