# Leaf multiplicity in a Bienaymé-Galton-Watson tree

@article{Brandenberger2021LeafMI, title={Leaf multiplicity in a Bienaym{\'e}-Galton-Watson tree}, author={Anna M. Brandenberger and Luc Devroye and Marcel K. Goh and Rosie Y. Zhao}, journal={Discret. Math. Theor. Comput. Sci.}, year={2021}, volume={24} }

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…

## References

SHOWING 1-10 OF 21 REFERENCES

### Local Limits of Large Galton-Watson Trees Rerooted at a Random Vertex

- MathematicsAofA
- 2018

In the condensation regime, in complete generality the asymptotic local behaviour from a random vertex up to its first ancestor with large degree is described, and in a subregime of complete condensation, convergence toward a novel limit tree is obtained.

### Asymptotic normality of fringe subtrees and additive functionals in conditioned Galton–Watson trees

- Mathematics, Computer ScienceRandom Struct. Algorithms
- 2016

This work considers conditioned Galton–Watson trees and shows asymptotic normality of additive functionals that are defined by toll functions that are not too large, including, as a special case, the number of fringe subtrees isomorphic to any given tree, and joint asymptic normality for several such subtrees.

### The Number of Trees

- Mathematics
- 1948

The mathematical theory of trees was first discussed by Cayley in 1857 (1). He was successful in finding recursion formulas for counting the number of trees or rooted trees having a finite number of…

### On the Height Profile of a Conditioned Galton-Watson Tree

- Mathematics
- 2011

Drmota and Gittenberger (1997) proved a conjecture due to Aldous (1991) on the height profile of a Galton-Watson tree with an offspring distribution of finite variance, conditioned on a total size of…

### Entropy Computations via Analytic Depoissonization

- Computer Science, MathematicsIEEE Trans. Inf. Theory
- 1999

It is argued that analytic methods can offer new tools for information theory, especially for studying second-order asymptotics, and there has been a resurgence of interest and a few successful applications of analytic methods to a variety of problems of information theory.

### Root estimation in Galton–Watson trees

- Mathematics, Computer ScienceRandom Struct. Algorithms
- 2022

The maximum‐likelihood estimator for the root of a free tree when the underlying tree is a size‐conditioned Galton–Watson tree is determined and its probability of being correct is calculated.

### Probability on Trees and Networks

- Mathematics
- 2017

Starting around the late 1950s, several research communities began relating the geometry of graphs to stochastic processes on these graphs. This book, twenty years in the making, ties together…

### Subdiffusive behavior of random walk on a random cluster

- Mathematics
- 1986

On considere deux cas de marche aleatoire {X n } n≥0 sur un graphe aleatoire #7B-G. Dans le cas ou #7B-G est l'arbre d'un processus de branchement critique, conditionne par la non-extinction, si h(x)…

### On the Altitude of Nodes in Random Trees

- MathematicsCanadian Journal of Mathematics
- 1978

Let Tn denote a tree with n nodes that is rooted at node r. (For definitions not given here see [4] or [10].) The altitude of a node u in Tn is the distance α = α (u, Tn) between r and u in Tn. The…

### On the Probability of the Extinction of Families

- History
- 1875

The decay of the families of men who occupied conspicuous positions in past times has been a subject of frequent remark, and has given rise to various conjectures. It is not only the families of men…