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13
Martingales and Profile of Binary Search Trees
, 2004
"... We are interested in the asymptotic analysis of the binary search tree (BST) under the random permutation model. Via an embedding in a continuous time model, we get new results, in particular the asymptotic behavior of the profile. ..."
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Cited by 48 (11 self)
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We are interested in the asymptotic analysis of the binary search tree (BST) under the random permutation model. Via an embedding in a continuous time model, we get new results, in particular the asymptotic behavior of the profile.
Profiles of random trees: Limit theorems for random recursive trees and binary search trees
, 2005
"... 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 Œ0; e/. Convergence of all moments is shown to hold only for ˛ 2 Œ0; 1 (with only con ..."
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Cited by 26 (11 self)
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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 Œ0; e/. Convergence of all moments is shown to hold only for ˛ 2 Œ0; 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.
A functional limit theorem for the profile of search trees
 ANNALS OF APPLIED PROBABILITY
, 2008
"... We study the profile Xn,k of random search trees including binary search trees and mary search trees. Our main result is a functional limit theorem of the normalized profile Xn,k/EXn,k for k =⌊α log n ⌋ in a certain range of α. A central feature of the proof is the use of the contraction method to ..."
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Cited by 26 (11 self)
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We study the profile Xn,k of random search trees including binary search trees and mary search trees. Our main result is a functional limit theorem of the normalized profile Xn,k/EXn,k for k =⌊α log n ⌋ 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 infinitedimensional Hilbert space. As part of the proof, we show that the Zolotarev metric is complete for a Hilbert space.
Profiles of random trees: correlation and width of random recursive trees and binary search trees
 ADVANCES IN APPLIED PROBABILITY
, 2004
"... We derive asymptotic approximations to the correlation coefficients of two level sizes in random recursive trees and binary search trees, which undergo sharp signchanges when one level is fixed and the other one is varying. An asymptotic estimate for the expected width is also derived. ..."
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Cited by 20 (7 self)
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We derive asymptotic approximations to the correlation coefficients of two level sizes in random recursive trees and binary search trees, which undergo sharp signchanges when one level is fixed and the other one is varying. An asymptotic estimate for the expected width is also derived.
The random multisection problem, travelling waves, and the distribution of the height of mary search trees
, 2006
"... The purpose of this article is to show that the distribution of the longest fragment in the random multisection problem after k steps and the height of mary search trees (and some extensions) are not only closely related in a formal way but both can be asymptotically described with the same distrib ..."
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Cited by 10 (2 self)
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The purpose of this article is to show that the distribution of the longest fragment in the random multisection problem after k steps and the height of mary search trees (and some extensions) are not only closely related in a formal way but both can be asymptotically described with the same distribution function that has to be shifted in a proper way (travelling wave). The crucial property for the proof is a socalled intersection property that transfers inequalities between two distribution functions (resp. of their Laplace transforms) from one level to the next. It is conjectured that such intersection properties hold in a much more general context. If this property is verified convergence to a travelling wave follows almost automatically.
THE FERMAT CUBIC, ELLIPTIC FUNCTIONS, CONTINUED FRACTIONS, AND A COMBINATORIAL EXCURSION
, 2005
"... Elliptic functions considered by Dixon in the nineteenth century and related to Fermat’s cubic, x 3 +y 3 = 1, lead to a new set of continued fraction expansions with sextic numerators and cubic denominators. The functions and the fractions are pregnant with interesting combinatorics, including a s ..."
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Cited by 7 (2 self)
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Elliptic functions considered by Dixon in the nineteenth century and related to Fermat’s cubic, x 3 +y 3 = 1, lead to a new set of continued fraction expansions with sextic numerators and cubic denominators. The functions and the fractions are pregnant with interesting combinatorics, including a special Pólya urn, a continuoustime branching process of the Yule type, as well as permutations satisfying various constraints that involve either parity of levels of elements or a repetitive pattern of order three. The combinatorial models are related to but different from models of elliptic functions earlier introduced by Viennot, Flajolet, Dumont, and Françon.