  The Horton-Strahler number, also known as the register function, provides a
tool for quantifying the branching complexity of a rooted tree. We consider the
Horton-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 the
result in Brandenberger, Devroye \& Reddad [6] dealing with the finite variance
case for which $\alpha=1$. We also characterize the cases where $\alpha=1$,
namely the spectrally positive Cauchy regime, which exhibits more complex
behaviors.
