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56
Random trees and applications
, 2005
"... We discuss several connections between discrete and continuous ..."
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Cited by 29 (4 self)
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We discuss several connections between discrete and continuous
Random cutting and records in deterministic and random trees
 Alg
, 2006
"... Abstract. We study random cutting down of a rooted tree and show that the number of cuts is equal (in distribution) to the number of records in the tree when edges (or vertices) are assigned random labels. Limit theorems are given for this number, in particular when the tree is a random conditioned ..."
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Cited by 29 (9 self)
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Abstract. We study random cutting down of a rooted tree and show that the number of cuts is equal (in distribution) to the number of records in the tree when edges (or vertices) are assigned random labels. Limit theorems are given for this number, in particular when the tree is a random conditioned Galton–Watson tree. We consider both the distribution when both the tree and the cutting (or labels) are random, and the case when we condition on the tree. The proofs are based on Aldous ’ theory of the continuum random tree. 1.
Parking Functions, Empirical Processes, and the Width of Rooted Labeled Trees
"... This paper provides tight bounds for the moments of the width of rooted labeled trees with n nodes, answering an open question of Odlyzko and Wilf (1987). To this aim, we use one of the many onetoone correspondences between trees and parking functions, and also a precise coupling between parking f ..."
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Cited by 20 (5 self)
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This paper provides tight bounds for the moments of the width of rooted labeled trees with n nodes, answering an open question of Odlyzko and Wilf (1987). To this aim, we use one of the many onetoone correspondences between trees and parking functions, and also a precise coupling between parking functions and the empirical processes of mathematical statistics. Our result turns out to be a consequence of the strong convergence of empirical processes to the Brownian bridge (Komlos, Major and Tusnady, 1975).
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 17 (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.
The density of the ISE and local limit laws for embedded trees
 Ann. Appl. Probab
"... Abstract. It has been known for a few years that the occupation measure of several models of embedded trees converges, after a suitable normalization, to the random measure called ISE (Integrated SuperBrownian Excursion). Here, we prove a local version of this result: ISE has a (random) Hölder conti ..."
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Cited by 16 (6 self)
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Abstract. It has been known for a few years that the occupation measure of several models of embedded trees converges, after a suitable normalization, to the random measure called ISE (Integrated SuperBrownian Excursion). Here, we prove a local version of this result: ISE has a (random) Hölder continuous density, and the vertical profile of embedded trees converges to this density, at least for some such trees. As a consequence, we derive a formula for the distribution of the density of ISE at a given point. This follows from earlier results by BousquetMélou on convergence of the vertical profile at a fixed point. We also provide a recurrence relation defining the moments of the (random) moments of ISE. 1.
The brownian excursion multidimensional local time density
 Journal of Applied Probability
, 1999
"... Expressions for the multidimensional densities of Brownian excursion local time are derived by two different methods: A direct method based on Kac’s formula for Brownian functionals and an indirect one based on a limit theorem for GaltonWatson trees. ..."
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Cited by 11 (9 self)
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Expressions for the multidimensional densities of Brownian excursion local time are derived by two different methods: A direct method based on Kac’s formula for Brownian functionals and an indirect one based on a limit theorem for GaltonWatson trees.
Stochastic Analysis Of TreeLike Data Structures
 Proc. R. Soc. Lond. A
, 2002
"... The purpose of this article is to present two types of data structures, binary search trees and usual (combinatorial) binary trees. Although they constitute the same set of (rooted) trees they are constructed via completely dierent rules and thus the underlying probabilitic models are dierent, too. ..."
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Cited by 9 (1 self)
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The purpose of this article is to present two types of data structures, binary search trees and usual (combinatorial) binary trees. Although they constitute the same set of (rooted) trees they are constructed via completely dierent rules and thus the underlying probabilitic models are dierent, too. Both kinds of data structures can be analyzed by probabilistic and stochastic tools, binary search trees (more or less) with martingales and binary trees (which can be considered as a special case of GaltonWatson trees) with stochastic processes. It is also an aim of this article to demonstrate the strength of analytic methods in speci c parts of probabilty theory related to combinatorial problems, especially we make use of the concept of generating functions. One reason is that that recursive combinatorial descriptions can be translated to relations for generating functions, and second analytic properties of these generating functions can be used to derive asymptotic (probabilistic) relations. 1.
The Width of GaltonWatson Trees Conditioned by the Size
, 2004
"... It is proved that the moments of the width of GaltonWatson trees of size n and with ospring variance are asymptotically given by ( n) mp where mp are the moments of the maximum of the local time of a standard scaled Brownian excursion. This is done by combining a weak limit theorem and a ti ..."
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Cited by 7 (1 self)
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It is proved that the moments of the width of GaltonWatson trees of size n and with ospring variance are asymptotically given by ( n) mp where mp are the moments of the maximum of the local time of a standard scaled Brownian excursion. This is done by combining a weak limit theorem and a tightness estimate. The method is quite general and we state some further applications. 1.
The connectivityprofile of random increasing ktrees
"... Random increasing ktrees represent an interesting, useful class of strongly dependent graphs for which analyticcombinatorial tools can be successfully applied. We study in this paper a notion called connectivityprofile and derive asymptotic estimates for it; some interesting consequences will als ..."
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Cited by 7 (2 self)
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Random increasing ktrees represent an interesting, useful class of strongly dependent graphs for which analyticcombinatorial tools can be successfully applied. We study in this paper a notion called connectivityprofile and derive asymptotic estimates for it; some interesting consequences will also be given. 1