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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.
Mixed Poisson approximation of node depth distributions in random binary search trees, Annals of Applied Probability 15
, 2005
"... We investigate the distribution of the depth of a node containing a specific key or, equivalently, the number of steps needed to retrieve an item stored in a randomly grown binary search tree. Using a representation in terms of mixed and compounded standard distributions, we derive approximations by ..."
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Cited by 5 (1 self)
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We investigate the distribution of the depth of a node containing a specific key or, equivalently, the number of steps needed to retrieve an item stored in a randomly grown binary search tree. Using a representation in terms of mixed and compounded standard distributions, we derive approximations by Poisson and mixed Poisson distributions; these lead to asymptotic normality results. We are particularly interested in the influence of the key value on the distribution of the node depth. Methodologically our message is that the explicit representation may provide additional insight if compared to the standard approach that is based on the recursive structure of the trees. Further, in order to exhibit the influence of the key on the distributional asymptotics, a suitable choice of distance of probability distributions is important. Our results are also applicable in connection with the number of recursions needed in Hoare’s [Comm. ACM 4 (1961) 321–322] selection algorithm Find. 1. Introduction. The
Analysis of some statistics for increasing tree families
, 2004
"... This paper deals with statistics concerning distances between randomly chosen nodes in varieties of increasing trees. Increasing trees are labelled rooted trees where labels along any branch from the root go in increasing order. Many important tree families that have applications in computer science ..."
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Cited by 4 (0 self)
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This paper deals with statistics concerning distances between randomly chosen nodes in varieties of increasing trees. Increasing trees are labelled rooted trees where labels along any branch from the root go in increasing order. Many important tree families that have applications in computer science or are used as probabilistic models in various applications, like recursive trees, heapordered trees or binary increasing trees (isomorphic to binary search trees) are members of this variety of trees. We consider the parameters depth of a randomly chosen node, distance between two randomly chosen nodes, and the generalisations where p nodes are randomly chosen: the size of the ancestortree of these selected nodes and the size of the smallest subtree generated by these nodes, also called Steiner distance. Under the restriction that the nodedegrees are bounded, we can prove that all these parameters converge in law to the Normal distribution. This extends results obtained earlier for binary search trees and heapordered trees to a much larger class of structures.
Extremal Weighted Path Lengths in Random Binary Search Trees
, 2008
"... Abstract. We consider weighted path lengths to the extremal leaves in a random binary search tree. When linearly scaled, the weighted path length to the minimal label has Dickman’s infinitely divisible distribution as a limit. By contrast, the weighted path length to the maximal label needs to be ce ..."
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Cited by 3 (0 self)
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Abstract. We consider weighted path lengths to the extremal leaves in a random binary search tree. When linearly scaled, the weighted path length to the minimal label has Dickman’s infinitely divisible distribution as a limit. By contrast, the weighted path length to the maximal label needs to be centered and scaled to converge to a standard normal variate in distribution. The exercise shows that path lengths associated with different ranks exhibit different behaviors depending on the rank. However, the majority of the ranks have a weighted path length with average behavior similar to that of the weighted path to the maximal node.
Distances in random digital search trees
, 2006
"... Distances between nodes in random trees is a popular topic, and several classes of trees have recently been investigated. We look into this matter in digital search trees. By analytic techniques, such as the Mellin Transform and poissonization, we describe a program to determine the moments of the ..."
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Cited by 2 (1 self)
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Distances between nodes in random trees is a popular topic, and several classes of trees have recently been investigated. We look into this matter in digital search trees. By analytic techniques, such as the Mellin Transform and poissonization, we describe a program to determine the moments of these distances. The program is illustrated on the mean and variance. One encounters delayed Mellin transform equations, which we solve by inspection. In addition to various asymptotics, we give an exact expression for the mean and for the variance in the unbiased case. Interestingly, the unbiased case gives a bounded variance, whereas the biased case gives a variance growing with the number of keys. It is therefore possible in the biased case to show that an appropriately normalized version of the distance converges to a limit. The complexity of moment calculation increases substantially with each higher moment; it is prudent to seek a shortcut to the limit via a method that avoids the computation of all moments. Toward this end, we utilize the contraction method to show that in biased digital search trees the distribution of a suitably normalized version of the distances approaches a limit that is the fixedpoint solution of a distributional equation (distances being measured in the Wasserstein metric space). An explicit solution to the fixedpoint equation is readily demonstrated to be Gaussian.
Applications of Steins method in the analysis of random binary search trees. Steins method and Applications
 Institute for Mathematical Sciences Lecture Notes Series
, 2005
"... Abstract. Under certain conditions, sums of functions of subtrees of a random binary search tree are asymptotically normal. We show how Stein’s method can be applied to study these random trees, and survey methods for obtaining limit laws for such functions of subtrees. ..."
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Cited by 1 (0 self)
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Abstract. Under certain conditions, sums of functions of subtrees of a random binary search tree are asymptotically normal. We show how Stein’s method can be applied to study these random trees, and survey methods for obtaining limit laws for such functions of subtrees.
The leftrightimbalance of binary search trees
 Theoretical Computer Science
, 2007
"... Abstract. We present a detailed study of leftrightimbalance measures for random binary search trees under the random permutation model, i.e., where binary search trees are generated by random permutations of {1, 2,..., n}. For random binary search trees of size n we study (i) the difference betwee ..."
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Cited by 1 (0 self)
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Abstract. We present a detailed study of leftrightimbalance measures for random binary search trees under the random permutation model, i.e., where binary search trees are generated by random permutations of {1, 2,..., n}. For random binary search trees of size n we study (i) the difference between the left and the right depth of a randomly chosen node, (ii) the difference between the left and the right depth of a specified node j = j(n), and (iii) the difference between the left and the right pathlength, and show for all three imbalance measures limiting distribution results. 1.
Distribution of internode distances in digital trees
 in 2005 International Conference on Analysis of Algorithms, C. Martínez (ed.), Discrete Mathematics and Theoretical Computer Science, Proceedings AD
, 2005
"... We investigate distances between pairs of nodes in digital trees (digital search trees (DST), and tries). By analytic techniques, such as the Mellin Transform and poissonization, we describe a program to determine the moments of these distances. The program is illustrated on the mean and variance. O ..."
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Cited by 1 (0 self)
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We investigate distances between pairs of nodes in digital trees (digital search trees (DST), and tries). By analytic techniques, such as the Mellin Transform and poissonization, we describe a program to determine the moments of these distances. The program is illustrated on the mean and variance. One encounters delayed Mellin transform equations, which we solve by inspection. Interestingly, the unbiased case gives a bounded variance, whereas the biased case gives a variance growing with the number of keys. It is therefore possible in the biased case to show that an appropriately normalized version of the distance converges to a limit. The complexity of moment calculation increases substantially with each higher moment; A shortcut to the limit is needed via a method that avoids the computation of all moments. Toward this end, we utilize the contraction method to show that in biased digital search trees the distribution of a suitably normalized version of the distances approaches a limit that is the fixedpoint solution (in the Wasserstein space) of a distributional equation. An explicit solution to the fixedpoint equation is readily demonstrated to be Gaussian.
The Wiener Index of Random Digital Trees
, 2012
"... The Wiener index has been studied for simply generated random trees, nonplane unlabeled random trees and a huge subclass of random grid trees containing random binary search trees, random medianof(2k + 1) search trees, random mary search trees, random quadtrees, random simplex trees, etc. An impo ..."
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The Wiener index has been studied for simply generated random trees, nonplane unlabeled random trees and a huge subclass of random grid trees containing random binary search trees, random medianof(2k + 1) search trees, random mary search trees, random quadtrees, random simplex trees, etc. An important class of random trees for which the Wiener index was not studied so far are random digital trees. In this work, we close this gap. More precisely, we derive asymptotic expansions of moments of the Wiener index and show that a central limit law for the Wiener index holds. These results are obtained for digital search trees and bucket versions as well as tries and PATRICIA tries. Our findings answer in affirmative two questions posed by Neininger. 1