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Limit laws for functions of fringe trees for binary search trees and random recursive trees.
Stockholm University, Faculty of Science, Department of Mathematics.
Uppsala universitet, Sweden.
2015 (English)In: Electronic Journal of Probability, ISSN 1083-6489, E-ISSN 1083-6489, Vol. 20, 4Article in journal (Refereed) Published
Abstract [en]

We prove general limit theorems for sums of functions of subtrees of (random) binary search trees and random recursive trees. The proofs use a new version of a representation by Devroye, and Stein's method for both normal and Poisson approximation together with certain couplings.As a consequence, we give simple new proofs of the fact that the number of fringe trees of size k=kn in the binary search tree or in the random recursive tree  (of total size n) has an asymptotical Poisson distribution if k→∞, and that the distribution is asymptotically normal for k=o(n√). Furthermore, we prove similar results for the number of subtrees of size k with some required property P, e.g., the number of copies of a certain fixed subtree T. Using the Cramér-Wold device, we show also that these random numbers for different fixed subtrees converge jointly to a multivariate normal distribution. We complete the paper by giving examples of applications of the general results, e.g., we obtain a normal limit law for the number of ℓ-protected nodes in a binary search tree or in a random recursive tree.

Place, publisher, year, edition, pages
2015. Vol. 20, 4
Keyword [en]
Fringe trees; Stein's method; Couplings; Limit laws; Binary search trees; Random recursive trees
National Category
Mathematics
Identifiers
URN: urn:nbn:se:su:diva-114312DOI: 10.1214/EJP.v20-3627ISI: 000350286000001OAI: oai:DiVA.org:su-114312DiVA: diva2:791147
Available from: 2015-02-26 Created: 2015-02-26 Last updated: 2017-12-04Bibliographically approved

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