Profiles of random trees: Limit theorems for random recursive trees and binary search trees

Michael Fuchs*, Hsien Kuei Hwang, Ralph Neininger

*Corresponding author for this work

Research output: Contribution to journalArticle

32 Scopus citations


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.

Original languageEnglish
Pages (from-to)367-407
Number of pages41
JournalAlgorithmica (New York)
Issue number3-4
StatePublished - 1 Nov 2006


  • Probabilistic analysis of algorithms
  • Profile of trees
  • Random binary search tree
  • Random recursive tree
  • Weak convergence

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