# Vertices with fixed outdegrees in large Galton-Watson trees

## TL;DR

This paper investigates the behavior of nodes with fixed outdegrees in large conditioned Galton-Watson trees, analyzing their scaling limits and asymptotic distributions across different exploration methods.

## Contribution

It extends existing results by providing necessary and sufficient conditions for the centeredness of the processes and generalizes the asymptotic normality of node counts.

## Key findings

- Derived conditions for centered limiting processes.
- Established asymptotic normality of node counts.
- Extended previous results to broader exploration methods.

## Abstract

We are interested in nodes with fixed outdegrees in large conditioned Galton--Watson trees. We first study the scaling limits of processes coding the evolution of the number of such nodes in different explorations of the tree (lexicographical order and contour order) starting from the root. We give necessary and sufficient conditions for the limiting processes to be centered, thus measuring the linearity defect of the evolution of the number of nodes with fixed outdegrees. This extends results by Labarbe & Marckert in the case of the contour-ordered counting process of leaves in uniform plane trees. Then, we extend results obtained by Janson concerning the asymptotic normality of the number of nodes with fixed outdegrees.