Multivariate Normal Limit Laws for the Numbers of Fringe Subtrees in $m$-ary Search Trees and Preferential Attachment Trees

Cecilia Holmgren, Svante Janson, Matas Sileikis


We study fringe subtrees of random $m$-ary search trees and of preferential attachment trees, by putting them in the context of generalised Pólya urns. In particular we show that for the random $m$-ary search trees with $ m\leq 26 $ and for the linear preferential attachment trees, the number of fringe subtrees that are isomorphic to an arbitrary fixed tree $ T $ converges to a normal distribution; more generally, we also prove multivariate normal distribution results for random vectors of such numbers for different fringe subtrees. Furthermore, we show that the number of protected nodes in random $m$-ary search trees for $m\leq 26$ has asymptotically a normal distribution.


Random trees; Fringe trees; Normal limit laws; Pólya urns; $m$-ary search trees; Preferential attachment trees; Protected nodes

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