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 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.

ON SEVERAL PROPERTIES OF A CLASS OF PREFERENTIAL ATTACHMENT TREES—PLANE-ORIENTED RECURSIVE TREES

2020 ◽  
pp. 1-19
Author(s):  
Panpan Zhang
Keyword(s):  
Continuous Time ◽  
Preferential Attachment ◽  
Mass Function ◽  
Probability Mass Function ◽  
Zagreb Index ◽  
Exact Probability ◽  
Probability Mass ◽  
Proper Scaling ◽  
Recursive Trees ◽  
Degree Variable

In this paper, several properties of a class of trees presenting preferential attachment phenomenon—plane-oriented recursive trees (PORTs) are uncovered. Specifically, we investigate the degree profile of a PORT by determining the exact probability mass function of the degree of a node with a fixed label. We compute the expectation and the variance of degree variable via a Pólya urn approach. In addition, we study a topological index, Zagreb index, of this class of trees. We calculate the exact first two moments of the Zagreb index (of PORTs) by using recurrence methods. Lastly, we determine the limiting degree distribution in PORTs that grow in continuous time, where the embedding is done in a Poissonization framework. We show that it is exponential after proper scaling.

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 Random Recursive Trees and Preferential Attachment Trees are Random Split Trees

2018 ◽  
Vol 28 (1) ◽  
pp. 81-99 ◽  
Author(s):  
SVANTE JANSON
Keyword(s):  
Preferential Attachment ◽  
Recursive Tree ◽  
The Common ◽  
Split Trees ◽  
Recursive Trees

We consider linear preferential attachment trees, and show that they can be regarded as random split trees in the sense of Devroye (1999), although with infinite potential branching. In particular, this applies to the random recursive tree and the standard preferential attachment tree. An application is given to the sum over all pairs of nodes of the common number of ancestors.

scholarly journals Embedding Small Digraphs and Permutations in Binary Trees and Split Trees

Algorithmica ◽  
2020 ◽  
Vol 82 (3) ◽  
pp. 589-615
Author(s):  
Michael Albert ◽  
Cecilia Holmgren ◽  
Tony Johansson ◽  
Fiona Skerman
Keyword(s):  
Large Class ◽  
Random Permutation ◽  
Random Trees ◽  
Binary Trees ◽  
Independent Interest ◽  
Explicit Formulas ◽  
Random Node ◽  
Labelled Trees ◽  
Logarithmic Height ◽  
Split Trees

AbstractWe investigate the number of permutations that occur in random labellings of trees. This is a generalisation of the number of subpermutations occurring in a random permutation. It also generalises some recent results on the number of inversions in randomly labelled trees (Cai et al. in Combin Probab Comput 28(3):335–364, 2019). We consider complete binary trees as well as random split trees a large class of random trees of logarithmic height introduced by Devroye (SIAM J Comput 28(2):409–432, 1998. 10.1137/s0097539795283954). Split trees consist of nodes (bags) which can contain balls and are generated by a random trickle down process of balls through the nodes. For complete binary trees we show that asymptotically the cumulants of the number of occurrences of a fixed permutation in the random node labelling have explicit formulas. Our other main theorem is to show that for a random split tree, with probability tending to one as the number of balls increases, the cumulants of the number of occurrences are asymptotically an explicit parameter of the split tree. For the proof of the second theorem we show some results on the number of embeddings of digraphs into split trees which may be of independent interest.

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 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 Large deviations for the leaves in some random trees

2009 ◽  
Vol 41 (3) ◽  
pp. 845-873 ◽  
Author(s):  
Wlodek Bryc ◽  
David Minda ◽  
Sunder Sethuraman
Keyword(s):  
Markov Chains ◽  
Large Deviations ◽  
Random Graphs ◽  
Large Deviation ◽  
Preferential Attachment ◽  
Random Trees ◽  
Explicit Computation ◽  
Large Deviation Principles ◽  
Recursive Trees

Large deviation principles and related results are given for a class of Markov chains associated to the ‘leaves' in random recursive trees and preferential attachment random graphs, as well as the ‘cherries’ in Yule trees. In particular, the method of proof, combining analytic and Dupuis–Ellis-type path arguments, allows for an explicit computation of the large deviation pressure.

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: 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
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