Abstract
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.
Highlights
We will use Polya urns to analyze vectors of the numbers of fringe subtrees of different types in random m-ary search trees and general preferential attachment trees, and in the former class we will analyze the number of protected nodes
We state the results on fringe subtrees and protected nodes in random m-ary search trees as well as fringe trees in preferential attachment trees
As said in the introduction, m-ary search trees can be regarded as either ordered or unordered trees; it is further possible to consider the labelled version as in Remark 2.2. (See [19, Remark 1.2] for the special case of the binary search tree.) The most natural interpretation is perhaps the one as ordered trees, and it immediately implies the corresponding result for unordered trees in, for example, Theorem 3.2
Summary
We will use (generalised) Polya urns to analyze vectors of the numbers of fringe subtrees of different types in random m-ary search trees and general (linear) preferential attachment trees, and in the former class we will analyze the number of protected nodes (that is, nodes with distance to a nearest leaf at least two). In particular we show that for the random m-ary search trees with m ≤ 26 and for the linear preferential attachment trees, the number of fringe subtrees that are isomorphic to an arbitrary fixed tree T has an asymptotic normal distribution; more generally, we prove multivariate normal distribution results for random vectors of such numbers for different trees.
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