The Shape of Unlabeled Rooted Random Trees

TL;DR

This paper studies the structure of unlabelled rooted random trees, showing their level sizes converge to Brownian excursion local time and analyzing their height distribution, extending known results for Galton-Watson trees.

Contribution

It extends the understanding of unlabelled rooted trees by establishing their asymptotic behavior and height distribution, similar to conditioned Galton-Watson processes.

Findings

Level sizes converge to Brownian excursion local time

Average height computed and characterized

Distribution of height derived

Abstract

We consider the number of nodes in the levels of unlabelled rooted random trees and show that the stochastic process given by the properly scaled level sizes weakly converges to the local time of a standard Brownian excursion. Furthermore we compute the average and the distribution of the height of such trees. These results extend existing results for conditioned Galton-Watson trees and forests to the case of unlabelled rooted trees and show that they behave in this respect essentially like a conditioned Galton-Watson process.