Fluctuations of the Horton–Strahler number of stable Galton–Watson trees

The Horton–Strahler number—also called the register function—is a combinatorial tool that quantifies the branching complexity of a rooted tree. We study the law of the Horton–Strahler number of the canonical stable Galton–Watson trees conditioned to have size n (including the Catalan trees), which are the finite-dimensional marginals of stable Lévy trees. While these random variables are known to grow as a multiple of lnn in probability, their fluctuations are not well understood because they are coupled with deterministic oscillations. To rule out the latter, we introduce a real-valued variant of the Horton–Strahler number. We show that a rescaled exponential of this quantity jointly converges in distribution to a measurable function of the scaling limit of the trees, that is, the stable Lévy tree. We call this limit the Strahler dilation, and we discuss its similarities with the Horton–Strahler number.