The Annals of Applied Probability

A functional limit theorem for the profile of search trees

Michael Drmota, Svante Janson, and Ralph Neininger

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We study the profile Xn,k of random search trees including binary search trees and m-ary search trees. Our main result is a functional limit theorem of the normalized profile $X_{n,k}/\mathbb{E}X_{n,k}$ for k=⌊αlogn⌋ in a certain range of α.

A central feature of the proof is the use of the contraction method to prove convergence in distribution of certain random analytic functions in a complex domain. This is based on a general theorem concerning the contraction method for random variables in an infinite-dimensional Hilbert space. As part of the proof, we show that the Zolotarev metric is complete for a Hilbert space.

Article information

Ann. Appl. Probab., Volume 18, Number 1 (2008), 288-333.

First available in Project Euclid: 9 January 2008

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Zentralblatt MATH identifier

Primary: 60F17: Functional limit theorems; invariance principles
Secondary: 68Q25: Analysis of algorithms and problem complexity [See also 68W40] 68P10: Searching and sorting 60C05: Combinatorial probability

Functional limit theorem search trees profile of trees random trees analysis of algorithms


Drmota, Michael; Janson, Svante; Neininger, Ralph. A functional limit theorem for the profile of search trees. Ann. Appl. Probab. 18 (2008), no. 1, 288--333. doi:10.1214/07-AAP457.

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