The Horton–Strahler number of conditioned Galton–Watson trees

The Horton–Strahler number of a tree is measure its branching complexity. It also known in the literature as register function. We show that for critical Galton–Watson trees with finite variance, conditioned to be size n, grows 1 2log2n probability. further define some generalizations this number. Among these are rigid and k-ary function, which we prove asymptotic results analogous standard case.