| Summary: | This note defines a notion of multiplicity for nodes in a rooted tree and presents an asymptotic calculation of the maximum multiplicity over all leaves in a Bienaymé-Galton-Watson tree with critical offspring distribution ξ, conditioned on the tree being of size n. In particular, we show that if Sn is the maximum multiplicity in a conditioned Bienaymé-Galton-Watson tree, then Sn = Ω(log n) asymptotically in probability and under the further assumption that E{2ξ} < ∞, we have Sn = O(log n) asymptotically in probability as well. Explicit formulas are given for the constants in both bounds. We conclude by discussing links with an alternate definition of multiplicity that arises in the root-estimation problem. © 2022 by the author(s).
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