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 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.
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.
"The Horton–Strahler number of conditioned Galton–Watson trees." Electron. J. Probab. 26 1 - 29, 2021. https://doi.org/10.1214/21-EJP678