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26
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.
Profile of Tries
, 2006
"... Tries (from retrieval) are one of the most popular data structures on words. They are pertinent to (internal) structure of stored words and several splitting procedures used in diverse contexts. The profile of a trie is a parameter that represents the number of nodes (either internal or external) wi ..."
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Cited by 21 (7 self)
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Tries (from retrieval) are one of the most popular data structures on words. They are pertinent to (internal) structure of stored words and several splitting procedures used in diverse contexts. The profile of a trie is a parameter that represents the number of nodes (either internal or external) with the same distance from the root. It is a function of the number of strings stored in a trie and the distance from the root. Several, if not all, trie parameters such as height, size, depth, shortest path, and fillup level can be uniformly analyzed through the (external and internal) profiles. Although profiles represent one of the most fundamental parameters of tries, they have been hardly studied in the past. The analysis of profiles is surprisingly arduous but once it is carried out it reveals unusually intriguing and interesting behavior. We present a detailed study of the distribution of the profiles in a trie built over random strings generated by a memoryless source. We first derive recurrences satisfied by the expected profiles and solve them asymptotically for all possible ranges of the distance from the root. It appears that profiles of tries exhibit several fascinating phenomena. When moving from the root to the leaves of a trie, the growth of the expected profiles vary. Near the root, the external profiles tend to zero in an exponentially rate, then the rate gradually rises to being logarithmic; the external profiles then abruptly tend to infinity, first logarithmically
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.
Profiles of random trees: planeoriented recursive trees
, 2005
"... We derive several limit results for the profile of random planeoriented recursive trees. These include the limit distribution of the normalized profile, asymptotic bimodality of the variance, asymptotic approximation to the expected width and the correlation coefficients of two level sizes. Most of ..."
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Cited by 17 (5 self)
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We derive several limit results for the profile of random planeoriented recursive trees. These include the limit distribution of the normalized profile, asymptotic bimodality of the variance, asymptotic approximation to the expected width and the correlation coefficients of two level sizes. Most of our proofs are based on a method of moments. We also discover an unexpected connection between the profile of planeoriented recursive trees (with logarithmic height) and that of random binary trees (with height proportional to the square root of tree size).
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.
Long and short paths in uniform random recursive dags
, 2009
"... In a uniform random recursive kdag, there is a root, 0, and each node in turn, from 1 to n, chooses k uniform random parents from among the nodes of smaller index. If Sn is the shortest path distance from node n to the root, then we determine the constant σ such that Sn / logn → σ in probability a ..."
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Cited by 8 (3 self)
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In a uniform random recursive kdag, there is a root, 0, and each node in turn, from 1 to n, chooses k uniform random parents from among the nodes of smaller index. If Sn is the shortest path distance from node n to the root, then we determine the constant σ such that Sn / logn → σ in probability as n → ∞. We also show that max1≤i≤n Si / logn → σ in probability.
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
On the subtree size profile of binary search trees
 Combin. Probab. Comput
"... Abstract. For random trees T generated by the binary search tree algorithm from uniformly distributed input we consider the subtree size profile, which maps k ∈ N to the number of nodes in T that root a subtree of size k. Complementing earlier work by Devroye, by Feng, Mahmoud and Panholzer, and by ..."
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Cited by 7 (3 self)
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Abstract. For random trees T generated by the binary search tree algorithm from uniformly distributed input we consider the subtree size profile, which maps k ∈ N to the number of nodes in T that root a subtree of size k. Complementing earlier work by Devroye, by Feng, Mahmoud and Panholzer, and by Fuchs, we obtain results for the range of small kvalues and the range of kvalues proportional to the size n of T. In both cases emphasis is on the process view, i.e. the joint distributions for several kvalues. We also show that the dynamics of the tree sequence lead to a qualitative difference between the asymptotic behaviour of the lower and the upper end of the profile. 1.
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.