Leaf multiplicity in a Bienaym\'e-Galton-Watson tree

This note defines a notion of multiplicity for nodes in a rooted tree andpresents an asymptotic calculation of the maximum multiplicity over all leavesin a Bienaym\'e-Galton-Watson tree with critical offspring distribution $\xi$,conditioned on the tree being of size $n$. In particular, we show that if $S_n$is the maximum multiplicity in a conditional Bienaym\'e-Galton-Watson tree,then $S_n = \Omega(\log n)$ asymptotically in probability and under the furtherassumption that ${\bf E}\{2^\xi\} < \infty$, we have $S_n = O(\log n)$asymptotically in probability as well. Explicit formulas are given for theconstants in both bounds. We conclude by discussing links with an alternatedefinition of multiplicity that arises in the root-estimation problem.Comment: 17 pages, 6 figures, final journal versio