Statistics of leaves in growing random trees

TL;DR

This paper investigates the leaf degree distribution in growing random trees, revealing factorial decay in recursive trees and diverse tail behaviors in a new leaf-based preferential attachment model, with implications for understanding sparse graph structures.

Contribution

It introduces a leaf-based attachment mechanism, analyzes its leaf degree distribution, and compares it with traditional degree distributions, providing new insights into tree growth dynamics.

Findings

Leaf degree distribution in recursive trees decays factorially.

The new model exhibits powerlaw, exponential, or stretched exponential tails depending on parameter a.

Conjecture of asymptotic equivalence between degree and leaf degree tail exponents.

Abstract

Leaves, i.e., vertices of degree one, can play a significant role in graph structure, especially in sparsely connected settings in which leaves often constitute the largest fraction of vertices. We consider a leaf-based counterpart of the degree, namely, the leaf degree -- the number of leaves a vertex is connected to -- and the associated leaf degree distribution, analogous to the degree distribution. We determine the leaf degree distribution of random recursive trees (RRTs) and trees grown via a leaf-based preferential attachment mechanism that we introduce. The RRT leaf degree distribution decays factorially, in contrast with its purely geometric degree distribution. In the one-parameter leaf-based growth model, each new vertex attaches to an existing vertex with rate $\ell$ + a, where $\ell$ is the leaf degree of the existing vertex, and a > 0. The leaf degree distribution has a powerlaw tail when 0 < a < 1 and an exponential tail (with algebraic prefactor) for a > 1. The critical case of a = 1 has a leaf degree distribution with stretched exponential tail. We compute a variety of additional characteristics in these models and conjecture asymptotic equivalence of degree and leaf degree powerlaw tail exponent in the scale free regime. We highlight several avenues of possible extension for future studies.