Depth properties of scaled attachment random recursive trees

TL;DR

This paper analyzes the depth characteristics of scaled attachment random recursive trees (SARRT), establishing asymptotic behaviors for their typical, maximum, and minimum depths, and providing a new proof for the height of uniform random recursive trees.

Contribution

It introduces SARRT, a general class of recursive trees, and derives their depth asymptotics, including a novel elementary proof for the height of uniform random recursive trees.

Findings

Typical depth D_n o rac{1}{ ext{μ}} imes ext{log} n

Maximum height H_n o ext{α}_{ ext{max}} imes ext{log} n

Minimum depth M_n o ext{α}_{ ext{min}} imes ext{log} n

Abstract

We study depth properties of a general class of random recursive trees where each node i attaches to the random node iX_i and X_0, ..., X_n is a sequence of i.i.d. random variables taking values in [0,1). We call such trees scaled attachment random recursive trees (SARRT). We prove that the typical depth D_n, the maximum depth (or height) H_n and the minimum depth M_n of a SARRT are asymptotically given by D_n \sim \mu^{-1} \log n, H_n \sim \alpha_{\max} \log n and M_n \sim \alpha_{\min} \log n where \mu, \alpha_{\max} and \alpha_{\min} are constants depending only on the distribution of X_0 whenever X_0 has a density. In particular, this gives a new elementary proof for the height of uniform random recursive trees H_n \sim e \log n that does not use branching random walks.