Leaf multiplicity in a Bienaym\'e-Galton-Watson tree

TL;DR

This paper introduces a new notion of node multiplicity in rooted trees and analyzes its maximum in critical Bienaymé-Galton-Watson trees, establishing asymptotic bounds and explicit constants.

Contribution

It provides the first asymptotic analysis of maximum leaf multiplicity in Bienaymé-Galton-Watson trees with explicit bounds and constants.

Findings

Maximum multiplicity grows at least logarithmically with tree size

Under finite exponential moment condition, maximum multiplicity is also bounded above by a logarithmic function

Explicit formulas for the constants in the asymptotic bounds are derived

Abstract

This note defines a notion of multiplicity for nodes in a rooted tree and presents an asymptotic calculation of the maximum multiplicity over all leaves in a Bienaym\'e-Galton-Watson tree with critical offspring distribution $\xi$, conditioned on the tree being of size $n$. In particular, we show that if $S_n$ is the maximum multiplicity in a conditional Bienaym\'e-Galton-Watson tree, then $S_n = \Omega(\log n)$ asymptotically in probability and under the further assumption that ${\bf E}\{2^\xi\} < \infty$, we have $S_n = O(\log n)$ asymptotically in probability as well. Explicit formulas are given for the constants in both bounds. We conclude by discussing links with an alternate definition of multiplicity that arises in the root-estimation problem.