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ALGORITHMICA
2006

Profiles of Random Trees: Limit Theorems for Random Recursive Trees and Binary Search Trees

13 years 11 months ago
Profiles of Random Trees: Limit Theorems for Random Recursive Trees and Binary Search Trees
We prove convergence in distribution for the profile (the number of nodes at each level), normalized by its mean, of random recursive trees when the limit ratio of the level and the logarithm of tree size lies in OE0; e/. Convergence of all moments is shown to hold only for 2 OE0; 1 (with only convergence of finite moments when 2 .1; e/). When the limit ratio is 0 or 1 for which the limit laws are both constant, we prove asymptotic normality for D 0 and a "quicksort type" limit law for D 1, the latter case having additionally a small range where there is no fixed limit law. Our tools are based on contraction method and method of moments. Similar phenomena also hold for other classes of trees; we apply our tools to binary search trees and give a complete characterization of the profile. The profiles of these random trees represent concrete examples for which the range of convergence in distribution differs from that of convergence of all moments.
Michael Fuchs, Hsien-Kuei Hwang, Ralph Neininger
Added 10 Dec 2010
Updated 10 Dec 2010
Type Journal
Year 2006
Where ALGORITHMICA
Authors Michael Fuchs, Hsien-Kuei Hwang, Ralph Neininger
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