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The Horton-Strahler number, also known as the register function, provides atool for quantifying the branching complexity of a rooted tree. We consider theHorton-Strahler number of critical Galton-Watson trees conditioned to have size$n$ and whose offspring distribution is in the domain of attraction of an$\alpha$-stable law with $\alpha\in [1, 2]$. We give tail estimates and when$\alpha\neq 1$, we prove that it grows as$\frac{1}{\alpha}\log_{\alpha/(\alpha-1)} n$ in probability. This extends theresult in Brandenberger, Devroye \& Reddad [6] dealing with the finite variancecase for which $\alpha=1$. We also characterize the cases where $\alpha=1$,namely the spectrally positive Cauchy regime, which exhibits more complexbehaviors.

The Horton-Strahler number of Galton-Watson trees with possibly infinite variance