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On the variance of the height of random binary search trees
 SIAM J
, 1995
"... Abstract. Let Hn be the height of a random binary search tree on n nodes. We show that there exist constants α = 4.311 ·· · and β = 1.953 ·· · such that E(Hn) = αln n − βln ln n + O(1), We also show that Var(Hn) = O(1). ..."
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Cited by 37 (3 self)
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Abstract. Let Hn be the height of a random binary search tree on n nodes. We show that there exist constants α = 4.311 ·· · and β = 1.953 ·· · such that E(Hn) = αln n − βln ln n + O(1), We also show that Var(Hn) = O(1).
TIGHTNESS FOR A FAMILY OF RECURSION EQUATIONS
"... Abstract. In this paper, we study the tightness of solutions for a family of recursion equations. These equations arise naturally in the study of random walks on treelike structures. Examples include the maximal displacement of branching random walk in one dimension, and the cover time of symmetric ..."
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Cited by 15 (3 self)
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Abstract. In this paper, we study the tightness of solutions for a family of recursion equations. These equations arise naturally in the study of random walks on treelike structures. Examples include the maximal displacement of branching random walk in one dimension, and the cover time of symmetric simple random walk on regular binary trees. Recursion equations associated with the distribution functions of these quantities have been used to establish weak laws of large numbers. Here, we use these recursion equations to establish the tightness of the corresponding sequences of distribution functions after appropriate centering. We phrase our results in a fairly general context, which we hope will facilitate their application in other settings. 1.
Minima in branching random walks
 Annals of Probability, 37(3): 1044–1079
, 2009
"... Given a branching random walk, let Mn be the minimum position of any member of the nth generation. We calculate EMn to within O(1) and prove exponential tail bounds for P{Mn −EMn > x}, under quite general conditions on the branching random walk. In particular, together with work by Bramson [Z. Wah ..."
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Cited by 13 (1 self)
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Given a branching random walk, let Mn be the minimum position of any member of the nth generation. We calculate EMn to within O(1) and prove exponential tail bounds for P{Mn −EMn > x}, under quite general conditions on the branching random walk. In particular, together with work by Bramson [Z. Wahrsch. Verw. Gebiete 45 (1978) 89–108], our results fully characterize the possible behavior of EMn when the branching random walk has bounded branching and step size. 1. Introduction. The
Connecting Yule process, bisection and binary search tree via martingales
 J. Iranian Statistical Society
, 2004
"... We present new links between some remarkable martingales found in the study of the Binary Search Tree or of the bisection problem, looking at them on the probability space of a continuous time binary branching process. Key words. Binary search tree, branching random walk, Yule process, convergence o ..."
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Cited by 9 (3 self)
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We present new links between some remarkable martingales found in the study of the Binary Search Tree or of the bisection problem, looking at them on the probability space of a continuous time binary branching process. Key words. Binary search tree, branching random walk, Yule process, convergence of martingales, functional equations. A.M.S. Classification. 60J25, 60J80, 60J85, 68W40, 60G42, 60G44. 1
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.
Profile and Height of Random Binary Search Trees
 JOURNAL OF THE IRANIAN STATISTICAL SOCIETY
, 2003
"... The purpose of this article is to survey recent results on distributional properties of random binary search trees. In particular we consider the profile and the height. ..."
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Cited by 7 (0 self)
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The purpose of this article is to survey recent results on distributional properties of random binary search trees. In particular we consider the profile and the height.
Tightness for the minimal displacement of branching random walk
, 2007
"... Abstract. Recursion equations have been used to establish weak laws of large numbers for the minimal displacement of branching random walk in one dimension. Here, we use these equations to establish the tightness of the corresponding sequences after appropriate centering. These equations are special ..."
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Cited by 7 (0 self)
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Abstract. Recursion equations have been used to establish weak laws of large numbers for the minimal displacement of branching random walk in one dimension. Here, we use these equations to establish the tightness of the corresponding sequences after appropriate centering. These equations are special cases of recursion equations that arise naturally in the study of random variables on treelike structures. Such recursion equations are investigated in detail, in [BZ06], in a general context. Here, we restrict ourselves to investigating the more concrete setting of branching random walk, and provide motivation for the rigorous arguments that are given in [BZ06]. We also discuss briefly the cover time of symmetric simple random walk on regular binary trees, which is another application of the more general recursion equations. 1.
The random multisection problem, travelling waves, and the distribution of the height of mary search trees
 Algorithmica
, 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 7 (1 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 Variance Of The Height Of Binary Search Trees
 Theoret. Comput. Sci
, 2000
"... . By using analytic tools it is shown that the variance E(Hn EHn ) 2 of the height Hn of binary search trees of size n is bounded. Keywords: binary search tree, average case analysis, generating functions 1. Introduction Let Hn denote the random variable which describes the height of binary searc ..."
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Cited by 3 (0 self)
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. By using analytic tools it is shown that the variance E(Hn EHn ) 2 of the height Hn of binary search trees of size n is bounded. Keywords: binary search tree, average case analysis, generating functions 1. Introduction Let Hn denote the random variable which describes the height of binary search trees of size n, i.e. P[Hn h] = a n;h n! ; where a n;h denotes the number of permutations 2 Sn of n elements such that the corresponding binary search tree has height h. (For a description of binary search trees see [6]. The height of a binary tree is the maximal distance of a leaf (external node) to the root.) In 1986 Devroye [1] proved that the expected value EHn satises the asymptotic relation EHn c log n (as n ! 1), where c = 4:31107 : : : is the solution greater than 2 of the equation 2e c c = e: (For related work and extensions see Devroye [2], Devroye and Reed [3], Pittel [7], and Robson [9, 10, 11].) Based on numerical data Robson conjectured that the variance VH...
Smoothed analysis of binary search trees and quicksort under additive noise
 Report 07039, Electronic Colloquium on Computational Complexity (ECCC
, 2007
"... Abstract. Binary search trees are a fundamental data structure and their height plays a key role in the analysis of divideandconquer algorithms like quicksort. We analyze their smoothed height under additive uniform noise: An adversary chooses a sequence of n real numbers in the range [0, 1], each ..."
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Cited by 3 (2 self)
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Abstract. Binary search trees are a fundamental data structure and their height plays a key role in the analysis of divideandconquer algorithms like quicksort. We analyze their smoothed height under additive uniform noise: An adversary chooses a sequence of n real numbers in the range [0, 1], each number is individually perturbed by adding a value drawn uniformly at random from an interval of size d, and the resulting numbers are inserted into a search tree. An analysis of the smoothed tree height subject to n and d lies at the heart of our paper: We prove that the smoothed height of binary search trees is Θ ( p n/d+log n), where d ≥ 1/n may depend on n. Our analysis starts with the simpler problem of determining the smoothed number of lefttoright maxima in a sequence. We establish matching bounds, namely once more Θ ( p n/d + log n). We also apply our findings to the performance of the quicksort algorithm and prove that the smoothed number of comparisons made by quicksort is Θ ( n d+1 p n/d + n log n). 1