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Robin Khanfir (McGill)

Published: 21 November 2023

The Horton-Strahler number and the Brownian tree


The Horton-Strahler number is an easy-to-compute integer that measures the branching complexity of a rooted tree. It has been first introduced by two hydrogeologists, Horton in 1945 and Strahler in 1952, to classify real-world river systems. However, the Horton-Strahler number has been rediscovered many times by virtually all scientific disciplines dealing with branching phenomena, such that molecular biology, anatomy, computer science, or even social network analysis. Like most previous works in probability theory, we focus on the law of the Horton-Strahler number of uniform binary trees with n leaves, the so-called Catalan trees. While these random variables are known to grow as log_2(n)/2, their fluctuations are not well-understood because they are coupled with deterministic oscillations. In this talk, we will discuss this asymptotic behavior and we will relate it to the scaling limit of Catalan trees, which is the celebrated Brownian Tree.

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