Open Access
2021 The Horton–Strahler number of conditioned Galton–Watson trees
Anna Brandenberger, Luc Devroye, Tommy Reddad
Author Affiliations +
Electron. J. Probab. 26: 1-29 (2021). DOI: 10.1214/21-EJP678


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 1 2log2n 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.

Funding Statement

Luc Devroye’s research was supported by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada (NSERC).


We are very grateful to the anonymous referees for their constructive comments, notably for an elegant proof of Lemma 2.1. We would also like to thank Konrad Anand, Marcel Goh, Jad Hamdan, Tyler Kastner, Gavin McCracken, Ndiamé Ndiaye, and Rosie Zhao for moral support, feedback and enlightening discussions. Special thanks to Marcel Goh for expert knowledge of the TE Xbook.


Download Citation

Anna Brandenberger. Luc Devroye. Tommy Reddad. "The Horton–Strahler number of conditioned Galton–Watson trees." Electron. J. Probab. 26 1 - 29, 2021.


Received: 17 October 2020; Accepted: 7 July 2021; Published: 2021
First available in Project Euclid: 20 July 2021

arXiv: 2010.08613
Digital Object Identifier: 10.1214/21-EJP678

Primary: 60C05
Secondary: 05C05 , 60F05 , 60J80

Keywords: branching processes , Galton–Watson trees , Horton–Strahler number , probabilistic analysis , Register function

Vol.26 • 2021
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