scholarly journals The profile of unlabeled trees

2005 ◽  
Vol DMTCS Proceedings vol. AD,... (Proceedings) ◽  
Author(s):  
Bernhard Gittenberger
Keyword(s):  
Local Time ◽  
Joint Distribution ◽  
Random Trees ◽  
Rooted Trees ◽  
Brownian Excursion ◽  
Level Number ◽  
International Audience ◽  
The Times ◽  
Simply Generated Trees

International audience We consider the number of nodes in the levels of unlabeled rooted random trees and show that the joint distribution of several level sizes (where the level number is scaled by $\sqrt{n}$) weakly converges to the distribution of the local time of a Brownian excursion evaluated at the times corresponding to the level numbers. This extends existing results for simply generated trees and forests to the case of unlabeled rooted trees.

scholarly journals The distance profile of rooted and unrooted simply generated trees

2021 ◽  
pp. 1-43
Author(s):  
Gabriel Berzunza Ojeda ◽  
Svante Janson
Keyword(s):  
Local Time ◽  
Random Function ◽  
Convergence Result ◽  
Rooted Trees ◽  
Height Profile ◽  
Hölder Continuous ◽  
Brownian Excursion ◽  
Open Questions ◽  
A Minor ◽  
Simply Generated Trees

Abstract 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.

scholarly journals Pattern distribution in various types of random trees

2005 ◽  
Vol DMTCS Proceedings vol. AD,... (Proceedings) ◽  
Author(s):  
Gerard Kok
Keyword(s):  
Normal Distribution ◽  
Limiting Distribution ◽  
Random Trees ◽  
Rooted Trees ◽  
Finite Tree ◽  
International Audience ◽  
Pattern Distribution ◽  
Simply Generated Trees

International audience Let $\mathcal{T}_n$ denote the set of unrooted unlabeled trees of size $n$ and let $\mathcal{M}$ be a particular (finite) tree. Assuming that every tree of $\mathcal{T}_n$ is equally likely, it is shown that the number of occurrences $X_n$ of $\mathcal{M}$ as an induced sub-tree satisfies $\mathbf{E} X_n \sim \mu n$ and $\mathbf{V}ar X_n \sim \sigma^2 n$ for some (computable) constants $\mu > 0$ and $\sigma \geq 0$. Furthermore, if $\sigma > 0$ then $(X_n - \mathbf{E} X_n) / \sqrt{\mathbf{V}ar X_n}$ converges to a limiting distribution with density $(A+Bt^2)e^{-Ct^2}$ for some constants $A,B,C$. However, in all cases in which we were able to calculate these constants, we obtained $B=0$ and thus a normal distribution. Further, if we consider planted or rooted trees instead of $T_n$ then the limiting distribution is always normal. Similar results can be proved for planar, labeled and simply generated trees.

scholarly journals On the distribution of the number of vertices in layers of random trees

1991 ◽  
Vol 4 (3) ◽  
pp. 175-186 ◽  
Author(s):  
Lajos Takács
Keyword(s):  
Asymptotic Behavior ◽  
Branching Processes ◽  
Random Trees ◽  
Rooted Trees ◽  
Brownian Excursion

Denote by Sn the set of all distinct rooted trees with n labeled vertices. A tree is chosen at random in the set Sn, assuming that all the possible nn−1 choices are equally probable. Define τn(m) as the number of vertices in layer m, that is, the number of vertices at a distance m from the root of the tree. The distance of a vertex from the root is the number of edges in the path from the vertex to the root. This paper is concerned with the distribution and the moments of τn(m) and their asymptotic behavior in the case where m=[2αn], 0<α<∞ and n→∞. In addition, more random trees, branching processes, the Bernoulli excursion and the Brownian excursion are also considered.

scholarly journals Simply generated trees, conditioned Galton―Watson trees, random allocations and condensation: Extended abstract

2012 ◽  
Vol DMTCS Proceedings vol. AQ,... (Proceedings) ◽  
Author(s):  
Svante Janson
Keyword(s):  
Random Trees ◽  
The Other ◽  
Random Allocation ◽  
Allocation Model ◽  
Standard Case ◽  
Infinite Degree ◽  
International Audience ◽  
Watson Process ◽  
Simply Generated Trees ◽  
High Degree

International audience We give a unified treatment of the limit, as the size tends to infinity, of random simply generated trees, including both the well-known result in the standard case of critical Galton-Watson trees and similar but less well-known results in the other cases (i.e., when no equivalent critical Galton-Watson tree exists). There is a well-defined limit in the form of an infinite random tree in all cases; for critical Galton-Watson trees this tree is locally finite but for the other cases the random limit has exactly one node of infinite degree. The random infinite limit tree can in all cases be constructed by a modified Galton-Watson process. In the standard case of a critical Galton-Watson tree, the limit tree has an infinite "spine", where the offspring distribution is size-biased. In the other cases, the spine has finite length and ends with a vertex with infinite degree. A node of infinite degree in the limit corresponds to the existence of one node with very high degree in the finite random trees; in physics terminology, this is a type of condensation. In simple cases, there is one node with a degree that is roughly a constant times the number of nodes, while all other degrees are much smaller; however, more complicated behaviour is also possible. The proofs use a well-known connection to a random allocation model that we call balls-in-boxes, and we prove corresponding results for this model.

scholarly journals On the number of transversals in random trees

2012 ◽  
Vol DMTCS Proceedings vol. AQ,... (Proceedings) ◽  
Author(s):  
Bernhard Gittenberger ◽  
Veronika Kraus
Keyword(s):  
Generating Functions ◽  
Singularity Analysis ◽  
Random Trees ◽  
Alternative Proof ◽  
Fixed Size ◽  
Random Subset ◽  
Polya Trees ◽  
Vertex Set ◽  
International Audience ◽  
Simply Generated Trees

International audience We study transversals in random trees with n vertices asymptotically as n tends to infinity. Our investigation treats the average number of transversals of fixed size, the size of a random transversal as well as the probability that a random subset of the vertex set of a tree is a transversal for the class of simply generated trees and for Pólya trees. The last parameter was already studied by Devroye for simply generated trees. We offer an alternative proof based on generating functions and singularity analysis and extend the result to Pólya trees.

scholarly journals The Width of Galton-Watson Trees Conditioned by the Size

2004 ◽  
Vol Vol. 6 no. 2 ◽  
Author(s):  
Michael Drmota ◽  
Bernhard Gittenberger
Keyword(s):  
Limit Theorem ◽  
Local Time ◽  
Weak Limit ◽  
Brownian Excursion ◽  
International Audience ◽  
Weak Limit Theorem

International audience It is proved that the moments of the width of Galton-Watson trees of size n and with offspring variance σ ^2 are asymptotically given by (σ √n)^pm_p where m_p are the moments of the maximum of the local time of a standard scaled Brownian excursion. This is done by combining a weak limit theorem and a tightness estimate. The method is quite general and we state some further applications.

scholarly journals Concentration Properties of Extremal Parameters in Random Discrete Structures

2006 ◽  
Vol DMTCS Proceedings vol. AG,... (Proceedings) ◽  
Author(s):  
Michael Drmota
Keyword(s):  
Maximum Degree ◽  
Random Trees ◽  
Height Distribution ◽  
Exponential Tail ◽  
Scale Free ◽  
Main Emphasis ◽  
Discrete Structures ◽  
International Audience ◽  
Concentration Properties

International audience The purpose of this survey is to present recent results concerning concentration properties of extremal parameters of random discrete structures. A main emphasis is placed on the height and maximum degree of several kinds of random trees. We also provide exponential tail estimates for the height distribution of scale-free trees.

scholarly journals A combinatorial non-commutative Hopf algebra of graphs

2014 ◽  
Vol Vol. 16 no. 1 (Combinatorics) ◽  
Author(s):  
Adrian Tanasa ◽  
Gerard Duchamp ◽  
Loïc Foissy ◽  
Nguyen Hoang-Nghia ◽  
Dominique Manchon
Keyword(s):  
Hopf Algebra ◽  
Rooted Trees ◽  
Quantum Field ◽  
International Audience ◽  
The One ◽  
Planar Rooted Trees

Combinatorics International audience A non-commutative, planar, Hopf algebra of planar rooted trees was defined independently by one of the authors in Foissy (2002) and by R. Holtkamp in Holtkamp (2003). In this paper we propose such a non-commutative Hopf algebra for graphs. In order to define a non-commutative product we use a quantum field theoretical (QFT) idea, namely the one of introducing discrete scales on each edge of the graph (which, within the QFT framework, corresponds to energy scales of the associated propagators). Finally, we analyze the associated quadri-coalgebra and codendrifrom structures.

scholarly journals Monitoring potential ionospheric changes caused by Van earthquake (Mw 7.2)

2018 ◽  
Author(s):  
Samed Inyurt ◽  
Selcuk Peker ◽  
Cetin Mekik
Keyword(s):  
Solar Activity ◽  
Local Time ◽  
Earthquake Prediction ◽  
Statistical Test ◽  
Van Earthquake ◽  
The North ◽  
The Times ◽  
Geomagnetic Activities

Abstract. Many scientists from different disciplines have studied earthquakes for many years. As a result of these studies, it has been proposed that some changes take place in the ionosphere layer before, during or after earthquakes, and the ionosphere should be monitored in earthquake prediction studies. This study investigates the changes in the ionosphere created by the earthquake with magnitude of Mw=7.2 in the northwest of the Lake Erçek which is located to the north of the province of Van in Turkey on 23 October 2011 and at 1.41 pm local time (−3 UT) with the epicenter of 38.75° N, 43.36° E using the TEC values obtained by the Global Ionosphere Models (GIM) created by IONOLAB-TEC and CODE. In order to see whether the ionospheric changes obtained by the study in question were caused by the earthquake or not, the ionospheric conditions were studied by utilizing indices providing information on solar and geomagnetic activities (F10.7 cm, Kp, Dst). One of the results of the statistical test on the TEC values obtained from the both models, positive and negative anomalies were obtained for the times before, on the day of and after the earthquake, and the reasons for these anomalies are discussed in detail in the last section of the study. As the ionospheric conditions in the analyzed days were highly variable, it was thought that the anomalies were caused by geomagnetic effects, solar activity and the earthquake.

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