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 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 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 Analysis of some statistics for increasing tree families

2004 ◽  
Vol Vol. 6 no. 2 ◽  
Author(s):  
Alois Panholzer ◽  
Helmut Prodinger
Keyword(s):  
Computer Science ◽  
Normal Distribution ◽  
Probabilistic Models ◽  
Binary Search ◽  
Rooted Trees ◽  
Binary Search Trees ◽  
Search Trees ◽  
Ordered Trees ◽  
International Audience

International audience This paper deals with statistics concerning distances between randomly chosen nodes in varieties of increasing trees. Increasing trees are labelled rooted trees where labels along any branch from the root go in increasing order. Many mportant tree families that have applications in computer science or are used as probabilistic models in various applications, like \emphrecursive trees, heap-ordered trees or \emphbinary increasing trees (isomorphic to binary search trees) are members of this variety of trees. We consider the parameters \textitdepth of a randomly chosen node, \textitdistance between two randomly chosen nodes, and the generalisations where \textitp nodes are randomly chosen Under the restriction that the node-degrees are bounded, we can prove that all these parameters converge in law to the Normal distribution. This extends results obtained earlier for binary search trees and heap-ordered trees to a much larger class of structures.

scholarly journals The Limiting Distribution for the Number of Symbol Comparisons Used by QuickSort is Nondegenerate (Extended Abstract)

2012 ◽  
Vol DMTCS Proceedings vol. AQ,... (Proceedings) ◽  
Author(s):  
Patrick Bindjeme ◽  
james Allen fill
Keyword(s):  
Large Class ◽  
Continuous Time ◽  
Random Variable ◽  
Limiting Distribution ◽  
International Audience ◽  
The Mean

International audience In a continuous-time setting, Fill (2012) proved, for a large class of probabilistic sources, that the number of symbol comparisons used by $\texttt{QuickSort}$, when centered by subtracting the mean and scaled by dividing by time, has a limiting distribution, but proved little about that limiting random variable $Y$—not even that it is nondegenerate. We establish the nondegeneracy of $Y$. The proof is perhaps surprisingly difficult.

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 Genealogy for supercritical branching processes

2006 ◽  
Vol 43 (04) ◽  
pp. 1066-1076 ◽  
Author(s):  
Andreas Nordvall Lagerås ◽  
Anders Martin-Löf
Keyword(s):  
Marginal Distribution ◽  
Branching Processes ◽  
Limiting Distribution ◽  
Random Trees ◽  
Limiting Distributions ◽  
Marginal Distributions ◽  
Special Cases ◽  
Supercritical Branching Processes

We study the genealogy of so-called immortal branching processes, i.e. branching processes where each individual upon death is replaced by at least one new individual, and conclude that their marginal distributions are compound geometric. The result also implies that the limiting distributions of properly scaled supercritical branching processes are compound geometric. We exemplify our results with an expression for the marginal distribution for a class of branching processes that have recently appeared in the theory of coalescent processes and continuous stable random trees. The limiting distribution can be expressed in terms of the Fox H-function, and in special cases by the Meijer G-function.

scholarly journals Exponential bounds and tails for additive random recursive sequences

2007 ◽  
Vol Vol. 9 no. 1 (Analysis of Algorithms) ◽  
Author(s):  
Ludger Rüschendorf ◽  
Eva-Maria Schopp
Keyword(s):  
Analysis Of Algorithms ◽  
Sufficient Conditions ◽  
Random Trees ◽  
Divide And Conquer ◽  
Recursive Sequences ◽  
Recursive Structures ◽  
International Audience ◽  
Exponential Bounds ◽  
The Moment

Analysis of Algorithms International audience Exponential bounds and tail estimates are derived for additive random recursive sequences, which typically arise as functionals of recursive structures, of random trees or in recursive algorithms. In particular they arise as parameters of divide and conquer type algorithms. We derive tail bounds from estimates of the Laplace transforms and of the moment sequences. For the proof we use some classical exponential bounds and some variants of the induction method. The paper generalizes results of Rösler (% \citeyearNPRoesler:91, % \citeyearNPRoesler:92) and % \citeNNeininger:05 on subgaussian tails to more general classes of additive random recursive sequences. It also gives sufficient conditions for tail bounds of the form \exp(-a t^p) which are based on a characterization of \citeNKasahara:78.

scholarly journals On the bialgebra of functional graphs and differential algebras

1997 ◽  
Vol Vol. 1 ◽  
Author(s):  
Maurice Ginocchio
Keyword(s):  
Dynamical Systems ◽  
Differential Operators ◽  
Discrete Dynamical Systems ◽  
Rooted Trees ◽  
Connected Components ◽  
Differential Algebras ◽  
Noncommutative Polynomials ◽  
International Audience

International audience We develop the bialgebraic structure based on the set of functional graphs, which generalize the case of the forests of rooted trees. We use noncommutative polynomials as generating monomials of the functional graphs, and we introduce circular and arborescent brackets in accordance with the decomposition in connected components of the graph of a mapping of \1, 2, \ldots, n\ in itself as in the frame of the discrete dynamical systems. We give applications fordifferential algebras and algebras of differential operators.

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