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.

Limit Theorems for Subtree Size Profiles of Increasing Trees

2012 ◽  
Vol 21 (3) ◽  
pp. 412-441 ◽  
Author(s):  
MICHAEL FUCHS
Keyword(s):  
Limit Theorems ◽  
Mean Value ◽  
Binary Search ◽  
Analytic Method ◽  
Binary Search Trees ◽  
Search Trees ◽  
Limit Laws ◽  
Special Cases ◽  
Recursive Trees ◽  
Complex Analytic Method

Simple families of increasing trees were introduced by Bergeron, Flajolet and Salvy. They include random binary search trees, random recursive trees and random plane-oriented recursive trees (PORTs) as important special cases. In this paper, we investigate the number of subtrees of size k on the fringe of some classes of increasing trees, namely generalized PORTs and d-ary increasing trees. We use a complex-analytic method to derive precise expansions of mean value and variance as well as a central limit theorem for fixed k. Moreover, we propose an elementary approach to derive limit laws when k is growing with n. Our results have consequences for the occurrence of pattern sizes on the fringe of increasing 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 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.

scholarly journals On the Subtree Size Profile of Binary Search trees

2010 ◽  
Vol 19 (4) ◽  
pp. 561-578 ◽  
Author(s):  
FLORIAN DENNERT ◽  
RUDOLF GRÜBEL
Keyword(s):  
Qualitative Difference ◽  
Search Tree ◽  
Binary Search ◽  
Random Trees ◽  
Binary Search Tree ◽  
Binary Search Trees ◽  
Search Trees ◽  
Joint Distributions ◽  
Size Profile ◽  
Process View

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 k-values and the range of k-values proportional to the size n of T. In both cases emphasis is on the process view, i.e., the joint distributions for several k-values. 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.

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