scholarly journals Embedding Small Digraphs and Permutations in Binary Trees and Split Trees

Algorithmica ◽  
2020 ◽  
Vol 82 (3) ◽  
pp. 589-615
Author(s):  
Michael Albert ◽  
Cecilia Holmgren ◽  
Tony Johansson ◽  
Fiona Skerman
Keyword(s):  
Large Class ◽  
Random Permutation ◽  
Random Trees ◽  
Binary Trees ◽  
Independent Interest ◽  
Explicit Formulas ◽  
Random Node ◽  
Labelled Trees ◽  
Logarithmic Height ◽  
Split Trees

AbstractWe investigate the number of permutations that occur in random labellings of trees. This is a generalisation of the number of subpermutations occurring in a random permutation. It also generalises some recent results on the number of inversions in randomly labelled trees (Cai et al. in Combin Probab Comput 28(3):335–364, 2019). We consider complete binary trees as well as random split trees a large class of random trees of logarithmic height introduced by Devroye (SIAM J Comput 28(2):409–432, 1998. 10.1137/s0097539795283954). Split trees consist of nodes (bags) which can contain balls and are generated by a random trickle down process of balls through the nodes. For complete binary trees we show that asymptotically the cumulants of the number of occurrences of a fixed permutation in the random node labelling have explicit formulas. Our other main theorem is to show that for a random split tree, with probability tending to one as the number of balls increases, the cumulants of the number of occurrences are asymptotically an explicit parameter of the split tree. For the proof of the second theorem we show some results on the number of embeddings of digraphs into split trees which may be of independent interest.

scholarly journals Tree limits and limits of random trees

2021 ◽  
pp. 1-45
Author(s):  
Svante Janson
Keyword(s):  
Continuous Time ◽  
Branching Process ◽  
Preferential Attachment ◽  
Binary Search ◽  
Random Trees ◽  
Ordered Trees ◽  
Labelled Trees ◽  
Split Trees ◽  
Recursive Trees ◽  
Simply Generated Trees

Abstract We explore the tree limits recently defined by Elek and Tardos. In particular, we find tree limits for many classes of random trees. We give general theorems for three classes of conditional Galton–Watson trees and simply generated trees, for split trees and generalized split trees (as defined here), and for trees defined by a continuous-time branching process. These general results include, for example, random labelled trees, ordered trees, random recursive trees, preferential attachment trees, and binary search trees.

scholarly journals The $k$-Cut Model in Deterministic and Random Trees

10.37236/9486 ◽  
2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Gabriel Berzunza ◽  
Xing Shi Cai ◽  
Cecilia Holmgren
Keyword(s):  
Random Trees ◽  
Convergence In Distribution ◽  
Scale Free ◽  
Offspring Distribution ◽  
Logarithmic Height ◽  
Split Trees ◽  
Recursive Trees ◽  
Cutting Model

The \(k\)-cut number of rooted graphs was introduced by Cai et al. as a generalization of the classical cutting model by Meir and Moon. In this paper, we show that all moments of the \(k\)-cut number of conditioned Galton-Watson trees converge after proper rescaling, which implies convergence in distribution to the same limit law regardless of the offspring distribution of the trees. This extends the result of Janson. Using the same method, we also show that the \(k\)-cut number of various random or deterministic trees of logarithmic height converges in probability to a constant after rescaling, such as random split-trees, uniform random recursive trees, and scale-free random trees. 

scholarly journals Profiles of random trees: plane-oriented recursive trees

2005 ◽  
Vol DMTCS Proceedings vol. AD,... (Proceedings) ◽  
Author(s):  
Hsien-Kuei Hwang
Keyword(s):  
Correlation Coefficients ◽  
Random Trees ◽  
Binary Trees ◽  
Asymptotic Approximations ◽  
Square Root ◽  
International Audience ◽  
Expected Width ◽  
Random Binary Trees ◽  
Logarithmic Height ◽  
Recursive Trees

International audience We summarize several limit results for the profile of random plane-oriented recursive trees. These include the limit distribution of the normalized profile, asymptotic bimodality of the variance, asymptotic approximations of the expected width and the correlation coefficients of two level sizes. We also unveil an unexpected connection between the profile of plane-oriented recursive trees (with logarithmic height) and that of random binary trees (with height proportional to the square root of tree size).

scholarly journals Enumerating alternating tree families

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
Markus Kuba ◽  
Alois Panholzer
Keyword(s):  
Nous Obtenons ◽  
Asymptotic Results ◽  
Root Node ◽  
Ordered Trees ◽  
Exponential Generating Function ◽  
Random Node ◽  
Enumeration Problems ◽  
International Audience ◽  
Nous Examinons ◽  
Labelled Trees

International audience We study two enumeration problems for $\textit{up-down alternating trees}$, i.e., rooted labelled trees $T$, where the labels $ v_1, v_2, v_3, \ldots$ on every path starting at the root of $T$ satisfy $v_1 < v_2 > v_3 < v_4 > \cdots$. First we consider various tree families of interest in combinatorics (such as unordered, ordered, $d$-ary and Motzkin trees) and study the number $T_n$ of different up-down alternating labelled trees of size $n$. We obtain for all tree families considered an implicit characterization of the exponential generating function $T(z)$ leading to asymptotic results of the coefficients $T_n$ for various tree families. Second we consider the particular family of up-down alternating labelled ordered trees and study the influence of such an alternating labelling to the average shape of the trees by analyzing the parameters $\textit{label of the root node}$, $\textit{degree of the root node}$ and $\textit{depth of a random node}$ in a random tree of size $n$. This leads to exact enumeration results and limiting distribution results. Nous étudions deux problèmes de dénombrement d'$\textit{arbres alternés haut-bas}$ : par définition, ce sont des arbres munis d'une racine et tels que, pour tout chemin partant de la racine, les valeurs $v_1,v_2,v_3,\ldots$ associées aux nœuds du chemin satisfont la chaîne d'inégalités $v_1 < v_2 > v_3 < v_4 > \cdots$. D'une part, nous considérons diverses familles d'arbres intéressantes du point de vue de l'analyse combinatoire (comme les arbres de Motzkin, les arbres non ordonnés, ordonnés et $d$-aires) et nous étudions pour chaque famille le nombre total $T_n$ d'arbres alternés haut-bas de taille $n$. Nous obtenons pour toutes les familles d'arbres considérées une caractérisation implicite de la fonction génératrice exponentielle $T(z)$. Cette caractérisation nous renseigne sur le comportement asymptotique des coefficients $T_n$ de plusieurs familles d'arbres. D'autre part, nous examinons le cas particulier de la famille des arbres ordonnés : nous étudions l'influence de l'étiquetage alterné haut-bas sur l'allure générale de ces arbres en analysant trois paramètres dans un arbre aléatoire (valeur de la racine, degré de la racine et profondeur d'un nœud aléatoire). Nous obtenons alors des résultats en terme de distribution limite, mais aussi de dénombrement exact.

scholarly journals Almost Giant Clusters for Percolation on Large Trees with Logarithmic Heights

2013 ◽  
Vol 50 (3) ◽  
pp. 603-611 ◽  
Author(s):  
Jean Bertoin
Keyword(s):  
Random Trees ◽  
Applied Probability ◽  
Bond Percolation ◽  
General Criterion ◽  
Asymptotic Results ◽  
Percolation Clusters ◽  
Large Trees ◽  
Logarithmic Height

This paper is based on works presented at the 2012 Applied Probability Trust Lecture in Sheffield; its main purpose is to survey the recent asymptotic results of Bertoin (2012a) and Bertoin and Uribe Bravo (2012b) about Bernoulli bond percolation on certain large random trees with logarithmic height. We also provide a general criterion for the existence of giant percolation clusters in large trees, which answers a question raised by David Croydon.

Tangent bundle RRT: A randomized algorithm for constrained motion planning

Robotica ◽  
2014 ◽  
Vol 34 (1) ◽  
pp. 202-225 ◽  
Author(s):  
Beobkyoon Kim ◽  
Terry Taewoong Um ◽  
Chansu Suh ◽  
F. C. Park
Keyword(s):  
Motion Planning ◽  
Configuration Space ◽  
Tangent Bundle ◽  
Numerical Experiments ◽  
Randomized Algorithm ◽  
Random Trees ◽  
Constrained Motion ◽  
Wide Range ◽  
Random Node ◽  
Planning Problems

SUMMARYThe Tangent Bundle Rapidly Exploring Random Tree (TB-RRT) is an algorithm for planning robot motions on curved configuration space manifolds, in which the key idea is to construct random trees not on the manifold itself, but on tangent bundle approximations to the manifold. Curvature-based methods are developed for constructing tangent bundle approximations, and procedures for random node generation and bidirectional tree extension are developed that significantly reduce the number of projections to the manifold. Extensive numerical experiments for a wide range of planning problems demonstrate the computational advantages of the TB-RRT algorithm over existing constrained path planning algorithms.

scholarly journals Random walks on random trees

1973 ◽  
Vol 15 (1) ◽  
pp. 42-53 ◽  
Author(s):  
J. W. Moon
Keyword(s):  
Random Walks ◽  
Random Trees ◽  
First Passage Times ◽  
General Reference ◽  
First Passage ◽  
Labelled Trees ◽  
Passage Times

Let T denote one of the nn−2 trees with n labelled nodes that is rooted at a given node x (see [6] or [8] as a general reference on trees). If i and j are any two nodes of T, we write i ∼ j if they are joined by an edge in T. We want to consider random walks on T; we assume that when we are at a node i of degree d the probability that we proceed to node j at the next step is di–1 if i ∼ j and zero otherwise. Our object here is to determine the first two moments of the first return and first passage times for random walks on T when T is a specific tree and when T is chosen at random from the set of all labelled trees with certain properties.

7.—Convolution of Vector Measures

1975 ◽  
Vol 73 ◽  
pp. 117-135 ◽  
Author(s):  
A. J. White
Keyword(s):  
Banach Algebra ◽  
Large Class ◽  
Independent Interest ◽  
Scalar Case ◽  
Algebra Structure ◽  
Semi Group ◽  
Locally Compact ◽  
Vector Measures ◽  
Borel Sets ◽  
Image Position

SynopsisIn this paper we consider the problem of defining a convolution, analogous to the classical convolution of scalar measures, of measures defined on the Borel sets of a locally compact semi-groupSand having values in a Banach algebra. Using the bilinearintegral introduced by Bartle we show that the formalism of the scalar case persists in situations of considerable generality so that the formulasuitably interpreted, gives a Banach algebra structure to a large class ofvalued measures defined onS.Themethods exploit the connection between vector measures and operators and involve some results of independent interest.

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