The degree profile of P\'olya trees

TL;DR

This paper studies the distribution of nodes of specific degrees at different levels in random Pólya trees, showing convergence to a Brownian excursion local time and analyzing joint degree distributions.

Contribution

It introduces a detailed analysis of degree-specific profiles in Pólya trees and establishes their convergence to a continuous stochastic process.

Findings

Normalized degree profiles converge to a Brownian excursion local time.

Joint distributions of nodes of different degrees are characterized.

The results extend understanding of structural properties of Pólya trees.

Abstract

We investigate the profile of random P\'olya trees of size $n$ when only nodes of degree $d$ are counted in each level. It is shown that, as in the case where all nodes contribute to the profile, the suitably normalized profile process converges weakly to a Brownian excursion local time. Moreover, we investigate the joint distribution of the number of nodes of degree $d_1$ and $d_2$ in the levels of the tree.