arXiv subjects

Anna M. Brandenberger

Publications and source records attributed to Anna M. Brandenberger.

3 recordsLinked to original sources

arXiv2022

Leaf multiplicity in a Bienaymé-Galton-Watson tree

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 Bienaymé-Galton-Watson tree with critical offspring distribution $ξ$, conditioned on the tree being of size $n$. In particular, we show that if $S_n$ is the maximum multiplicity in a conditional Bienaymé-Galton-Watson tree, then $S_n = Ω(\log n)$ asymptotically in probability and under the further assumption that ${\bf E}\{2^ξ\} < \infty$, 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.

math.PR↗

arXiv2021

Root estimation in Galton-Watson trees

Given only the free-tree structure of a tree, the root estimation problem asks if one can guess which of the free tree's nodes is the root of the original tree. We determine the maximum-likelihood estimator for the root of a free tree when the underlying tree is a size-conditioned Galton-Watson tree and calculate its probability of being correct.

math.PR↗

arXiv2020

The Horton-Strahler Number of Conditioned Galton-Watson Trees

The Horton-Strahler number of a tree is a measure of its branching complexity; it is also known in the literature as the register function. We show that for critical Galton-Watson trees with finite variance conditioned to be of size $n$, the Horton-Strahler number grows as $\frac{1}{2}\log_2 n$ in probability. We further define some generalizations of this number. Among these are the rigid Horton-Strahler number and the $k$-ary register function, for which we prove asymptotic results analogous to the standard case.

math.PR↗