scholarly journals On Tail Bounds for Random Recursive Trees

2012 ◽  
Vol 49 (02) ◽  
pp. 566-581 ◽  
Author(s):  
Götz Olaf Munsonius
Keyword(s):  
Analysis Of Algorithms ◽  
Wiener Index ◽  
Binary Search ◽  
Random Trees ◽  
Stochastic Domination ◽  
Lower Tail ◽  
Probabilistic Analysis Of Algorithms ◽  
The Laplace Transform ◽  
Recursive Trees ◽  
Tail Bound

We consider a multivariate distributional recursion of sum type, as arises in the probabilistic analysis of algorithms and random trees. We prove an upper tail bound for the solution using Chernoff's bounding technique by estimating the Laplace transform. The problem is traced back to the corresponding problem for binary search trees by stochastic domination. The result obtained is applied to the internal path length and Wiener index of randomb-ary recursive trees with weighted edges and random linear recursive trees. Finally, lower tail bounds for the Wiener index of these trees are given.

scholarly journals On Tail Bounds for Random Recursive Trees

2012 ◽  
Vol 49 (2) ◽  
pp. 566-581 ◽  
Author(s):  
Götz Olaf Munsonius
Keyword(s):  
Analysis Of Algorithms ◽  
Wiener Index ◽  
Binary Search ◽  
Random Trees ◽  
Stochastic Domination ◽  
Lower Tail ◽  
Probabilistic Analysis Of Algorithms ◽  
The Laplace Transform ◽  
Recursive Trees ◽  
Tail Bound

We consider a multivariate distributional recursion of sum type, as arises in the probabilistic analysis of algorithms and random trees. We prove an upper tail bound for the solution using Chernoff's bounding technique by estimating the Laplace transform. The problem is traced back to the corresponding problem for binary search trees by stochastic domination. The result obtained is applied to the internal path length and Wiener index of random b-ary recursive trees with weighted edges and random linear recursive trees. Finally, lower tail bounds for the Wiener index of these trees are given.

scholarly journals The Wiener Index of Random Trees

2002 ◽  
Vol 11 (6) ◽  
pp. 587-597 ◽  
Author(s):  
RALPH NEININGER
Keyword(s):  
Path Length ◽  
Probabilistic Models ◽  
Wiener Index ◽  
Binary Search ◽  
Random Trees ◽  
Binary Search Trees ◽  
Search Trees ◽  
Limit Laws ◽  
Recursive Trees ◽  
Asymptotic Correlations

The Wiener index is analysed for random recursive trees and random binary search trees in uniform probabilistic models. We obtain expectations, asymptotics for the variances, and limit laws for this parameter. The limit distributions are characterized as the projections of bivariate measures that satisfy certain fixed point equations. Covariances, asymptotic correlations, and bivariate limit laws for the Wiener index and the internal path length are given.

scholarly journals Profiles of random trees: correlation and width of random recursive trees and binary search trees

2005 ◽  
Vol 37 (02) ◽  
pp. 321-341 ◽  
Author(s):  
Michael Drmota ◽  
Hsien-Kuei Hwang
Keyword(s):  
Asymptotic Estimate ◽  
Correlation Coefficients ◽  
Singularity Analysis ◽  
Binary Search ◽  
Random Trees ◽  
The Other ◽  
Binary Search Trees ◽  
Search Trees ◽  
Expected Width ◽  
Recursive Trees

In a tree, a level consists of all those nodes that are the same distance from the root. We derive asymptotic approximations to the correlation coefficients of two level sizes in random recursive trees and binary search trees. These coefficients undergo sharp sign-changes when one level is fixed and the other is varying. We also propose a new means of deriving an asymptotic estimate for the expected width, which is the number of nodes at the most abundant level. Crucial to our methods of proof is the uniformity achieved by singularity analysis.

scholarly journals Limit Theorems for Depths and Distances in Weighted Random B-Ary Recursive Trees

2011 ◽  
Vol 48 (4) ◽  
pp. 1060-1080 ◽  
Author(s):  
Götz Olaf Munsonius ◽  
Ludger Rüschendorf
Keyword(s):  
Path Length ◽  
Limit Theorems ◽  
Wiener Index ◽  
Random Trees ◽  
Random Node ◽  
Recursive Trees

Limit theorems are established for some functionals of the distances between two nodes in weighted random b-ary recursive trees. We consider the depth of the nth node and of a random node, the distance between two random nodes, the internal path length, and the Wiener index. As an application, these limit results imply, by an imbedding argument, corresponding limit theorems for further classes of random trees: plane-oriented recursive trees and random linear recursive trees.

scholarly journals The existence of a giant cluster for percolation on large Crump–Mode–Jagers trees

2020 ◽  
Vol 52 (1) ◽  
pp. 266-290
Author(s):  
G. Berzunza
Keyword(s):  
Preferential Attachment ◽  
Percolation Cluster ◽  
Binary Search ◽  
Random Trees ◽  
Binary Search Trees ◽  
Search Trees ◽  
The Family ◽  
Giant Cluster ◽  
Recursive Trees ◽  
Fragmentation Trees

AbstractIn this paper we consider random trees associated with the genealogy of Crump–Mode–Jagers processes and perform Bernoulli bond-percolation whose parameter depends on the size of the tree. Our purpose is to show the existence of a giant percolation cluster for appropriate regimes as the size grows. We stress that the family trees of Crump–Mode–Jagers processes include random recursive trees, preferential attachment trees, binary search trees for which this question has been answered by Bertoin [7], as well as (more general) m-ary search trees, fragmentation trees, and median-of-( $2\ell+1$ ) binary search trees, to name a few, where to our knowledge percolation has not yet been studied.

scholarly journals Tree limits and limits of random trees

2021 ◽  
pp. 1-45
Author(s):  
Svante Janson
Keyword(s):  
Continuous Time ◽  
Branching Process ◽  
Preferential Attachment ◽  
Binary Search ◽  
Random Trees ◽  
Ordered Trees ◽  
Labelled Trees ◽  
Split Trees ◽  
Recursive Trees ◽  
Simply Generated Trees

Abstract We explore the tree limits recently defined by Elek and Tardos. In particular, we find tree limits for many classes of random trees. We give general theorems for three classes of conditional Galton–Watson trees and simply generated trees, for split trees and generalized split trees (as defined here), and for trees defined by a continuous-time branching process. These general results include, for example, random labelled trees, ordered trees, random recursive trees, preferential attachment trees, and binary search trees.

scholarly journals Limit Theorems for Depths and Distances in Weighted Random B-Ary Recursive Trees

2011 ◽  
Vol 48 (04) ◽  
pp. 1060-1080 ◽  
Author(s):  
Götz Olaf Munsonius ◽  
Ludger Rüschendorf
Keyword(s):  
Path Length ◽  
Limit Theorems ◽  
Wiener Index ◽  
Random Trees ◽  
Random Node ◽  
Recursive Trees

Limit theorems are established for some functionals of the distances between two nodes in weighted randomb-ary recursive trees. We consider the depth of thenth node and of a random node, the distance between two random nodes, the internal path length, and the Wiener index. As an application, these limit results imply, by an imbedding argument, corresponding limit theorems for further classes of random trees: plane-oriented recursive trees and random linear recursive trees.

scholarly journals Profiles of Random Trees: Limit Theorems for Random Recursive Trees and Binary Search Trees

Algorithmica ◽  
2006 ◽  
Vol 46 (3-4) ◽  
pp. 367-407 ◽  
Author(s):  
Michael Fuchs ◽  
Hsien-Kuei Hwang ◽  
Ralph Neininger
Keyword(s):  
Limit Theorems ◽  
Binary Search ◽  
Random Trees ◽  
Binary Search Trees ◽  
Search Trees ◽  
Recursive Trees

scholarly journals Profiles of random trees: correlation and width of random recursive trees and binary search trees

2005 ◽  
Vol 37 (2) ◽  
pp. 321-341 ◽  
Author(s):  
Michael Drmota ◽  
Hsien-Kuei Hwang
Keyword(s):  
Asymptotic Estimate ◽  
Correlation Coefficients ◽  
Singularity Analysis ◽  
Binary Search ◽  
Random Trees ◽  
The Other ◽  
Binary Search Trees ◽  
Search Trees ◽  
Expected Width ◽  
Recursive Trees

In a tree, a level consists of all those nodes that are the same distance from the root. We derive asymptotic approximations to the correlation coefficients of two level sizes in random recursive trees and binary search trees. These coefficients undergo sharp sign-changes when one level is fixed and the other is varying. We also propose a new means of deriving an asymptotic estimate for the expected width, which is the number of nodes at the most abundant level. Crucial to our methods of proof is the uniformity achieved by singularity analysis.

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