  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. 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 L\'evy tree. We call this limit the
Strahler dilation and we discuss its similarities with the Horton-Strahler
number.
