Simply Generated Unrooted Plane Trees

Ramzews, Leon; Stufler, Benedikt

doi:10.30757/alea.v16-12

Cited by 2 publications

(1 citation statement)

References 36 publications

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“…A bicolored plane tree is a tree whose vertices are colored consecutively black and white. Generally each plane tree is related to two bicolored plane trees [1,3,17]. The distance between two distinct vertices of the tree T and the path length in accordance (with the number of edges) connecting is defined and denoted by d( ) [4,5].…”

Section: Introductionmentioning

confidence: 99%

Correspondence between two bicolored plane trees And generalized Tchebytchev polynomial

Hagri¹

2020

BUJ

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In this paper, we give some theoretical results, for the Tchebytchev polynomials of a tree. A Tchebytchev polynomial, any polynomial has at most two critical values. we consider two bicolored plane trees (two neighboring vertices are always indifferent colorings).We will introduce the composition and decomposition of two bicolored plane trees A and B denoted by (A o B) and we are interested in the relation between two bicolored plane trees and generalized Tchebytchev polynomials.

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Section: Introductionmentioning

confidence: 99%

Correspondence between two bicolored plane trees And generalized Tchebytchev polynomial

Hagri¹

2020

BUJ

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The distance profile of rooted and unrooted simply generated trees

Ojeda

Janson

2021

Combinator. Probab. Comp.

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It is well known that the height profile of a critical conditioned Galton–Watson tree with finite offspring variance converges, after a suitable normalisation, to the local time of a standard Brownian excursion. In this work, we study the distance profile, defined as the profile of all distances between pairs of vertices. We show that after a proper rescaling the distance profile converges to a continuous random function that can be described as the density of distances between random points in the Brownian continuum random tree. We show that this limiting function a.s. is Hölder continuous of any order $\alpha<1$ , and that it is a.e. differentiable. We note that it cannot be differentiable at 0, but leave as open questions whether it is Lipschitz, and whether it is continuously differentiable on the half-line $(0,\infty)$ . The distance profile is naturally defined also for unrooted trees contrary to the height profile that is designed for rooted trees. This is used in our proof, and we prove the corresponding convergence result for the distance profile of random unrooted simply generated trees. As a minor purpose of the present work, we also formalize the notion of unrooted simply generated trees and include some simple results relating them to rooted simply generated trees, which might be of independent interest.

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Simply Generated Unrooted Plane Trees

Cited by 2 publications

References 36 publications

Correspondence between two bicolored plane trees And generalized Tchebytchev polynomial

Correspondence between two bicolored plane trees And generalized Tchebytchev polynomial

The distance profile of rooted and unrooted simply generated trees

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