Scaling limit of trees with vertices of fixed degrees and heights

TL;DR

This paper investigates the asymptotic behavior of large uniform random trees with fixed vertex degrees and heights, establishing their convergence under certain conditions and applying the results to Bienaymé-Galton-Watson trees in varying environments.

Contribution

It introduces a convergence framework for these trees using coalescent processes and extends the results to Bienaymé-Galton-Watson trees with varying environments.

Findings

Proves convergence of scaled trees under profile convergence conditions

Analyzes paths from vertices to root via coalescent processes

Derives scaling limits for Bienaymé-Galton-Watson trees in varying environments

Abstract

We consider large uniform random trees where we fix for each vertex its degree and height. We prove, under natural conditions of convergence for the profile, that those trees properly renormalized converge. To this end, we study the paths from random vertices to the root using coalescent processes. As an application, we obtain scaling limits of Bienaym\'e-Galton-Watson trees in varying environment.