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

Authors:
Michael Fuchs;Hsien-Kuei Hwang;Ralph Neininger
Affiliations:
Department of Applied Mathematics, National Chiao Tung University, Hsinchu 300, Taiwan;Institute of Statistical Science, Academia Sinica, Taipei 115, Austria;Department of Mathematics, J.W. Goethe University, 60325, Frankfurt a.M., Germany
Venue:
Algorithmica
Year:
2006

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Abstract

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 α of the level and the logarithm of tree size lies in [0,e). Convergence of all moments is shown to hold only for α ∈ [0,1] (with only convergence of finite moments when α ∈ (1,e)). When the limit ratio is 0 or 1 for which the limit laws are both constant, we prove asymptotic normality for α = 0 and a "quicksort type" limit law for α = 1, the latter case having additionally a small range where there is no fixed limit law. Our tools are based on the 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.