Multivariate Normal Limit Laws for the Numbers of Fringe Subtrees in $m$-ary Search Trees and Preferential Attachment Trees

Holmgren, Cecilia; Janson, Svante; Šileikis, Matas

doi:10.37236/6374

Cited by 15 publications

(28 citation statements)

References 34 publications

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“…Also (1.3) holds, at least under some supplementary assumptions, see [8,Remark 4.2], and then the results above hold. (In the examples from [14] and [7] just mentioned, (1.3) holds because there is an equivalent urn with random replacements that satisfies the conditions of [8].) Remark 1.9.…”

Section: Introductionmentioning

confidence: 99%

Moment convergence of balanced Pólya processes

Janson¹,

Pouyanne²

2018

Electron. J. Probab.

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Section: Introductionmentioning

confidence: 99%

Moment convergence of balanced Pólya processes

Janson¹,

Pouyanne²

2018

Electron. J. Probab.

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“…[55] and [66]. For m-ary search trees, the situation is more complicated: no results for general fringe trees have been published (this is work in progress [68]), but some special cases (such as the degree distribution and the number of fringe trees of a given size) and related quantities (the number of internal nodes) have been treated, and it turns out that central limit theorems hold for m 26 but not for m 27, see e.g. [95], [93], [90], [30], [69], [29], [53], [74] and [67].…”

Section: Asymptotic Normality?mentioning

confidence: 99%

Fringe trees, Crump–Mode–Jagers branching processes and $m$-ary search trees

Holmgren¹,

Janson²

2017

Probab. Surveys

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This survey studies asymptotics of random fringe trees and extended fringe trees in random trees that can be constructed as family trees of a Crump-Mode-Jagers branching process, stopped at a suitable time. This includes random recursive trees, preferential attachment trees, fragmentation trees, binary search trees and (more generally) mary search trees, as well as some other classes of random trees.We begin with general results, mainly due to Aldous (1991) and Jagers and Nerman (1984). The general results are applied to fringe trees and extended fringe trees for several particular types of random trees, where the theory is developed in detail. In particular, we consider fringe trees of m-ary search trees in detail; this seems to be new.Various applications are given, including degree distribution, protected nodes and maximal clades for various types of random trees. Again, we emphasise results for m-ary search trees, and give for example new results on protected nodes in m-ary search trees.A separate section surveys results on height, saturation level, typical depth and total path length, due to Devroye (1986), Biggins (1995, 1997 and others.This survey contains well-known basic results together with some additional general results as well as many new examples and applications for various classes of random trees.

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“…In their very recent paper [10], Holmgren, Janson and Šileikis approached the same question from a different angle: using Pólya urns, they proved that the number of fringe subtrees isomorphic to a given rooted tree is asymptotically normally distributed. They also proved joint normality for different fringe subtrees.…”

Section: Introductionmentioning

confidence: 99%

A central limit theorem for additive functionals of increasing trees

Ralaivaosaona

Wagner

2019

Combinator. Probab. Comp.

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A tree functional is called additive if it satisfies a recursion of the form F (T ) = k j=1 F (Bj) + f (T ), where B1, . . . , B k are the branches of the tree T and f (T ) is a toll function. We prove a general central limit theorem for additive functionals of d-ary increasing trees under suitable assumptions on the toll function. The same method also applies to generalised plane-oriented increasing trees (GPORTs). One of our main applications is a log-normal law that we prove for the size of the automorphism group of d-ary increasing trees, but many other examples (old and new) are covered as well.

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Multivariate Normal Limit Laws for the Numbers of Fringe Subtrees in $m$-ary Search Trees and Preferential Attachment Trees

Cited by 15 publications

References 34 publications

Moment convergence of balanced Pólya processes

Moment convergence of balanced Pólya processes

Fringe trees, Crump–Mode–Jagers branching processes and $m$-ary search trees

A central limit theorem for additive functionals of increasing trees

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