Local limits of galton-watson trees conditioned on the number of protected nodes

TL;DR

This paper studies the asymptotic behavior of critical Galton-Watson trees conditioned on the number of marked or protected nodes, showing convergence to a size-biased tree and providing distributional limits.

Contribution

It introduces a new marking procedure dependent on out-degree and proves convergence results for conditioned Galton-Watson trees, including those conditioned on protected nodes.

Findings

Convergence of conditioned trees to size-biased trees

Distributional limits for trees with many protected nodes

Application of marking procedure to analyze node types

Abstract

We consider a marking procedure of the vertices of a tree where each vertex is marked independently from the others with a probability that depends only on its out-degree. We prove that a critical Galton-Watson tree conditioned on having a large number of marked vertices converges in distribution to the associated size-biased tree. We then apply this result to give the limit in distribution of a critical Galton-Watson tree conditioned on having a large number of protected nodes.