Vertices of degree <i>k</i> in random unlabeled trees

Παναγιώτου, Κωνσταντίνος; Sinha, Makrand

doi:10.1002/jgt.20567

Cited by 5 publications

(3 citation statements)

References 12 publications

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“…For example, [43] determines (among other more general things) g n with this technique in the prominent setting where c n is subexponential, which corresponds to our setting with α < 0. For other applications of the Pólya-Boltzmann model regarding random multisets, see for instance [40,39].…”

Section: Notes On the Proof And Perspectivementioning

confidence: 99%

Expansive Multisets: Asymptotic Enumeration

Παναγιώτου¹,

Ramzews²

2022

Preprint

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Consider a non-negative sequence cn = h(n) • n α−1 • ρ −n , where h is slowly varying, α > 0, 0 < ρ < 1 and n ∈ N. We investigate the coefficients of G(x, y) = k≥1 (1 − x k y) −c k , which is the bivariate generating series of the multiset construction of combinatorial objects. By a powerful blend of probabilistic methods based on the Boltzmann model and analytic techniques exploiting the well-known saddle-point method we determine the number of multisets of total size n with N components, that is, the coefficient of x n y N in G(x, y), asymptotically as n → ∞ and for all ranges of N . Our results reveal a phase transition in the structure of the counting formula that depends on the ratio n/N and that demonstrates a prototypical passage from a bivariate local limit to an univariate one.

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Section: Notes On the Proof And Perspectivementioning

confidence: 99%

Expansive Multisets: Asymptotic Enumeration

Παναγιώτου¹,

Ramzews²

2022

Preprint

Self Cite

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“…One of the earliest investigations of this kind was [42], where the enumeration of given stars as subgraphs in trees (equivalently nodes of fixed degree) was treated. Later generalizations are found in [11,40] (multivariate setting), in [35] (distinct patterns) or [24] (large patterns of that type). A method to deal with general contiguous patterns in trees by means of generating functions was developed in [6], which was partially generalized to planar maps recently [14,7,13].…”

Section: Introductionmentioning

confidence: 99%

Counting Embeddings of Rooted Trees into Families of Rooted Trees

Sułkowska

Gittenberger

Gołębiewski

et al. 2022

Electron. J. Combin.

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An embedding of a rooted tree $S$ into another rooted tree $T$ is a mapping of the vertex set of $S$ into the vertex set of $T$ that is order-preserving in a certain sense, depending on the chosen tree family. Equivalently, $S$ and $T$ may be interpreted as tree-like partially ordered sets, where $S$ is isomorphic to a subposets of $T$. A good embedding is an embedding where the root of $S$ is mapped to the root of $T$. We investigate the number of good and the number of all embeddings of a rooted tree $S$ into the family of all trees of given size $n$ of a certain family of trees. The tree families considered are binary plane trees (the order of descendants matters), binary non-plane trees and rooted plane trees. We derive the asymptotic behaviour of the number of good and the number of all embeddings in all these cases and prove that the ratio of the number of good embeddings to that of all embeddings is of the order $\Theta(1/\sqrt{n})$ in all cases, where we provide the exact constants as well. Furthermore, we prove some monotonicity properties of this ratio. Finally, we comment on the case where $S$ is disconnected.

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“…One of the earliest investigations of this kind was [38], where the enumeration of given stars as subgraphs in trees (equivalently nodes of fixed degree) was treated. Later generalizations are found in [10,36] (multivariate setting), in [33] (distinct patterns) or [22] (large patterns of that type). A method to deal with general contiguous patterns in trees by means of generating functions was developed in [5], which was partially generalized to planar maps recently [12,6,11].…”

Section: Introductionmentioning

confidence: 99%

Counting embeddings of rooted trees into families of rooted trees

Gittenberger¹,

Gołębiewski²,

Larcher³

et al. 2020

Preprint

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The number of embeddings of a partially ordered set S in a partially ordered set T is the number of subposets of T isomorphic to S. If both, S and T , have only one unique maximal element, we define good embeddings as those in which the maximal elements of S and T overlap. We investigate the number of good and all embeddings of a rooted poset S in the family of all binary trees on n elements considering two cases: plane (when the order of descendants matters) and non-plane. Furthermore, we study the number of embeddings of a rooted poset S in the family of all planted plane trees of size n. We derive the asymptotic behaviour of good and all embeddings in all cases and we prove that the ratio of good embeddings to all is of the order Θ(1/ √ n) in all cases, where we provide the exact constants. Furthermore, we show that this ratio is non-decreasing with S in the plane binary case and asymptotically non-decreasing with S in the non-plane binary case and in the planted plane case. Finally, we comment on the case when S is disconnected.

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Vertices of degree k in random unlabeled trees

Cited by 5 publications

References 12 publications

Expansive Multisets: Asymptotic Enumeration

Expansive Multisets: Asymptotic Enumeration

Counting Embeddings of Rooted Trees into Families of Rooted Trees

Counting embeddings of rooted trees into families of rooted trees

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