Degree distribution in the lower levels of the uniform recursive tree

TL;DR

This paper analyzes the degree distribution in the lower levels of uniform recursive trees, showing convergence results and Poisson asymptotics for node degrees.

Contribution

It provides new asymptotic results on degree distributions in the lower levels of uniform recursive trees, including convergence and independence properties.

Findings

Proportion of nodes with degree > t log n converges to (1-t)^k almost surely.

Number of degree d nodes in first level is asymptotically Poisson with mean 1.

Degree counts are asymptotically independent for different degrees.

Abstract

In this note we consider the $k$th level of the uniform random recursive tree after $n$ steps, and prove that the proportion of nodes with degree greater than $t\log n$ converges to $(1-t)^k$ almost surely, as $n\to\infty$, for every $t\in(0,1)$. In addition, we show that the number of degree $d$ nodes in the first level is asymptotically Poisson distributed with mean 1; moreover, they are asymptotically independent for $d=1,2,...$.