NDLI: Phase Changes in Subtree Varieties in Random Recursive and Binary Search Trees

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Panholzer, Alois Mahmoud, Hosam M. Feng, Qunqiang
Copyright Year	2008
Abstract	We study the variety of subtrees lying on the fringe of recursive trees and binary search trees by analyzing the distributional behavior of $X_{n,k}$, which counts the number of subtrees of size k in a random tree of size n, with $k = k(n)$ dependent on n. Using analytic methods we can characterize for both tree families the phase change behavior of $X_{n,k}$ as follows. In the subcritical case, when $k(n)/\sqrt{n} \to 0$, we show that $X_{n,k}$ is (after normalization) asymptotically normally distributed, whereas in the supercritical case, when $k(n)/\sqrt n \to \infty$, $X_{n,k}$ converges to 0. In the critical case, when $k(n) = \Theta(\sqrt{n}\,)$, we show that if $k/\sqrt{n}$ approaches a limit, then $X_{n,k}$ converges in distribution to a Poisson random variable, whereas if $k/\sqrt{n}$ does not approach a finite nonzero limit, the size oscillates and does not converge in distribution to any random variable. In regard to recursive trees and binary search trees, this provides an understanding of the complete spectrum of phases of $X_{n,k}$ and the gradual change from the subcritical to the supercritical phase.
Starting Page	160
Ending Page	184
Page Count	25
File Format	PDF
ISSN	08954801
DOI	10.1137/060653950
e-ISSN	10957146
Journal	SIAM Journal on Discrete Mathematics (SJDMEC)
Issue Number	1
Volume Number	22
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2008-02-15
Access Restriction	Subscribed
Subject Keyword	recurrence Trees random trees moments Combinatorial probability Riccati equation Central limit and other weak theorems
Content Type	Text
Resource Type	Article
Subject	Mathematics

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