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48
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.
WIDTH AND MODE OF THE PROFILE FOR SOME RANDOM TREES OF LOGARITHMIC HEIGHT
 SUBMITTED TO THE ANNALS OF APPLIED PROBABILITY
, 2005
"... We propose a new, direct, correlationfree approach based on central moments of profiles to the asymptotics of width (size of the most abundant level) in some random trees of logarithmic height. The approach is simple but gives precise estimates for expected width, central moments of the width, and ..."
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Cited by 9 (1 self)
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We propose a new, direct, correlationfree approach based on central moments of profiles to the asymptotics of width (size of the most abundant level) in some random trees of logarithmic height. The approach is simple but gives precise estimates for expected width, central moments of the width, and almost sure convergence. It is widely applicable to random trees of logarithmic height, including recursive trees, binary search trees, quadtrees, planeoriented ordered trees and other varieties of increasing 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 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 8 (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
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.
The shape of unlabeled rooted random trees
, 2008
"... We consider the number of nodes in the levels of unlabeled rooted random trees and show that the stochastic process given by the properly scaled level sizes weakly converges to the local time of a standard Brownian excursion. Furthermore we compute the average and the distribution of the height of ..."
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Cited by 6 (3 self)
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We consider the number of nodes in the levels of unlabeled rooted random trees and show that the stochastic process given by the properly scaled level sizes weakly converges to the local time of a standard Brownian excursion. Furthermore we compute the average and the distribution of the height of such trees. These results extend existing results for conditioned GaltonWatson trees and forests to the case of unlabeled rooted trees and show that they behave in this respect essentially like a conditioned GaltonWatson process.
Congruence properties of depths in some random trees
, 2005
"... Consider a random recusive tree with n vertices. We show that the number of vertices with even depth is asymptotically normal as n → ∞. The same is true for the number of vertices of depth divisible by m for m = 3, 4 or 5; in all four cases the variance grows linearly. On the other hand, for m ≥ 7 ..."
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Cited by 5 (0 self)
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Consider a random recusive tree with n vertices. We show that the number of vertices with even depth is asymptotically normal as n → ∞. The same is true for the number of vertices of depth divisible by m for m = 3, 4 or 5; in all four cases the variance grows linearly. On the other hand, for m ≥ 7, the number is not asymptotically normal, and the variance grows faster than linear in n. The case m = 6 is intermediate: the number is asymptotically normal but the variance is of order nlog n. This is a simple and striking example of a type of phase transition that has been observed by other authors in several cases. We prove, and perhaps explain, this nonintuitive behavious using a translation to a generalized Pólya urn. Similar results hold for a random binary search tree; now the number of vertices of depth divisible by m is asymptotically normal for m ≤ 8 but not for m ≥ 9, and the variance grows linearly in the first case both faster in the second. (There is no intermediate case.) In contrast, we show that for conditioned Galton–Watson trees, including random labelled trees and random binary trees, there is no such phase transition: the number is asymptotically normal for every m.