A degree-biased cutting process for random recursive trees

TL;DR

This paper introduces a degree-biased cutting process on random recursive trees, deriving the distribution of deleted vertices, recursive formulas for the number of cuts, and connections to stable distributions and coalescing processes.

Contribution

It presents a novel degree-biased cutting process, explicit distributions, recursive formulas, and links to stable distributions and coalescent processes in recursive trees.

Findings

Distribution of vertices deleted per cut derived

Recursive formula for total cuts needed established

Convergence to a spectrally negative stable distribution

Abstract

We investigate a degree-biased cutting process on random recursive trees, where each vertex is deleted with probability proportional to its degree. We establish the splitting property and derive the explicit distribution of the number of vertices deleted in each cut. This leads to a recursive formula for Kn, the number of cuts needed to erase a random recursive tree with n vertices. Furthermore, we show that Kn is stochastically dominated by Jn, the number of jumps made by a related walk with a barrier. We prove that Jn converges in distribution to a random variable with a spectrally negative stable distribution. Finally, we examine connections between this cutting procedure and a coalescing process on the set of n elements.