Multivariate normal limit laws for the numbers of fringe subtrees in $ m $-ary search trees and preferential attachment trees

TL;DR

This paper establishes that the counts of specific fringe subtrees and protected nodes in certain random trees follow normal distributions asymptotically, using generalized Pólya urn models.

Contribution

It proves multivariate normal limit laws for fringe subtree counts and protected nodes in m-ary search trees and preferential attachment trees, extending previous results.

Findings

Counts of fringe subtrees are asymptotically normal.

Number of protected nodes converges to a normal distribution.

Results apply for m ≤ 26 in m-ary search trees.

Abstract

We study fringe subtrees of random $ m $-ary search trees and of preferential attachment trees, by putting them in the context of generalised P\'olya 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.