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 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 $\ln n$ 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, i.e. the stable Lévy tree. We call this limit the Strahler dilation and we discuss its similarities with the Horton-Strahler number.