  The Horton-Strahler number of a tree is a measure of its branching
complexity; it is also known in the literature as the register function. We
show that for critical Galton-Watson trees with finite variance conditioned to
be of size $n$, the Horton-Strahler number grows as $\frac{1}{2}\log_2 n$ in
probability. We further define some generalizations of this number. Among these
are the rigid Horton-Strahler number and the $k$-ary register function, for
which we prove asymptotic results analogous to the standard case.
