Limiting probabilities for vertices of a given rank in rooted trees

TL;DR

This paper investigates the limiting probabilities of vertices having specific ranks in two types of labeled rooted trees, demonstrating convergence of these probabilities as tree size increases.

Contribution

It establishes the convergence of vertex rank probabilities in labeled rooted trees as their size tends to infinity, providing new insights into their asymptotic structure.

Findings

Probabilities converge to a limit as tree size grows

Results apply to two varieties of labeled rooted trees

Provides a foundation for understanding vertex rank distribution

Abstract

We consider two varieties of labeled rooted trees, and the probability that a vertex chosen from all vertices of all trees of a given size uniformly at random has a given rank. We prove that this probability converges to a limit as the tree size goes to infinity.