Combinatorial Analysis of Growth Models for Series-Parallel Networks

Kuba, Markus; Panholzer, Alois

doi:10.1017/s096354831800038x

Cited by 2 publications

(3 citation statements)

References 21 publications

(85 reference statements)

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“…and φ k = d k , k 0, respectively. The three weight sequences (φ k ) k 0 together with (14) directly lead to the result of Theorem 1. Below we present the calculations for (b, α)-plane oriented recursive trees derived from generalized plane-oriented recursive trees with φ k = k+α−1 k .…”

Section: Algorithm 1 Clusteringincreasingtrees(t B)mentioning

confidence: 84%

“…Equations of this or similar kind have been treated in [15,10,14]. It follows from these considerations that, for κ given in (23), all solutions λ 1 , λ 2 , .…”

Section: Initial Bucket Size Of a Specified Elementmentioning

confidence: 99%

“…We analyzed structural properties of bucket increasing trees, in particular, the tree parameter K n , counting the initial bucket size of the node containing label n in a random tree of size n. Using the combinatorial description as well as the tree evolution process, a study of further quantities in bucket increasing tree families is possible, e.g., we want to mention node distances. Moreover, there exist relations of bucket increasing trees to further combinatorial structures as, e.g., certain models of series-parallel networks, see [14].…”

Section: Conclusion and Acknowledgmentsmentioning

confidence: 99%

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On bucket increasing trees, clustered increasing trees and increasing diamonds

Kuba¹,

Panholzer²

2020

Preprint

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In this work we analyze bucket increasing tree families. We introduce two simple stochastic growth processes, generating random bucket increasing trees of size n, complementing the earlier result of Mahmoud and Smythe [15] for bucket recursive trees. On the combinatorial side, we define multilabelled generalizations of the tree families d-ary increasing trees and generalized plane-oriented recursive trees. Additionally, we introduce a clustering process for ordinary increasing trees and relate it to bucket increasing trees. We discuss in detail the bucket size two and present a bijection between such bucket increasing tree families and certain families of graphs called increasing diamonds, providing an explanation for phenomena observed by Bodini et al. [3].Concerning structural properties of bucket increasing trees, we analyze the tree parameter K n . It counts the initial bucket size of the node containing label n in a tree of size n and is closely related to the distribution of node types. Additionally, we analyze the parameters descendants of label j and degree of the bucket containing label j, providing distributional decompositions, complementing and extending earlier results [10].

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Section: Algorithm 1 Clusteringincreasingtrees(t B)mentioning

confidence: 84%

“…Equations of this or similar kind have been treated in [15,10,14]. It follows from these considerations that, for κ given in (23), all solutions λ 1 , λ 2 , .…”

Section: Initial Bucket Size Of a Specified Elementmentioning

confidence: 99%

Section: Conclusion and Acknowledgmentsmentioning

confidence: 99%

See 1 more Smart Citation

On bucket increasing trees, clustered increasing trees and increasing diamonds

Kuba¹,

Panholzer²

2020

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

On bucket increasing trees, clustered increasing trees and increasing diamonds

Kuba

Panholzer

2021

Combinator. Probab. Comp.

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In this work we analyse bucket increasing tree families. We introduce two simple stochastic growth processes, generating random bucket increasing trees of size n, complementing the earlier result of Mahmoud and Smythe (1995, Theoret. Comput. Sci.144 221–249.) for bucket recursive trees. On the combinatorial side, we define multilabelled generalisations of the tree families d-ary increasing trees and generalised plane-oriented recursive trees. Additionally, we introduce a clustering process for ordinary increasing trees and relate it to bucket increasing trees. We discuss in detail the bucket size two and present a bijection between such bucket increasing tree families and certain families of graphs called increasing diamonds, providing an explanation for phenomena observed by Bodini et al. (2016, Lect. Notes Comput. Sci.9644 207–219.). Concerning structural properties of bucket increasing trees, we analyse the tree parameter $K_n$ . It counts the initial bucket size of the node containing label n in a tree of size n and is closely related to the distribution of node types. Additionally, we analyse the parameters descendants of label j and degree of the bucket containing label j, providing distributional decompositions, complementing and extending earlier results (Kuba and Panholzer (2010), Theoret. Comput. Sci.411(34–36) 3255–3273.).

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Combinatorial Analysis of Growth Models for Series-Parallel Networks

Cited by 2 publications

References 21 publications

On bucket increasing trees, clustered increasing trees and increasing diamonds

On bucket increasing trees, clustered increasing trees and increasing diamonds

On bucket increasing trees, clustered increasing trees and increasing diamonds

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