Large fringe trees for random trees with given vertex degrees

TL;DR

This paper investigates the asymptotic behavior of fringe trees in large random plane trees with specified degree distributions, extending prior normality results to growing target trees using multiple probabilistic methods.

Contribution

It introduces new asymptotic results for counts of growing fringe trees and compares four probabilistic frameworks to establish convergence conditions.

Findings

Identifies conditions for Poisson and normal convergence of fringe tree counts.

Provides a local limit theorem for sampling without replacement.

Applies results to conditioned critical Galton-Watson trees.

Abstract

This paper extends the study of fringe trees in random plane trees with a given degree statistic. While previous work established the asymptotic normality of the count of fringe trees isomorphic to a fixed tree, we investigate the case where the target tree grows with the size of the random tree. We consider three primary subtree counts: the number of fringe trees isomorphic to a specific growing tree, the number of fringe trees sharing a given growing degree statistic, and the number of fringe trees of a specific growing size. To establish our results, we employ and compare four distinct probabilistic frameworks: the method of moments with the Gao-Wormald theorem, Stein's method with coupling (to provide explicit error bounds in total variation distance), the Cai-Devroye method, and Stein's method with exchangeable pairs. Our findings provide conditions for Poisson and normal convergence for these subtree counts. Additionally, we provide a local limit theorem for sums of values obtained via sampling without replacement that may be of independent interest. Finally, our results and methods are also applied to conditioned critical Galton-Watson trees.