scholarly journals Descendants in Increasing Trees

10.37236/1034 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Markus Kuba ◽  
Alois Panholzer
Keyword(s):  
Probability Distribution ◽  
Limiting Distribution ◽  
Random Tree ◽  
Factorial Moments ◽  
Recursive Trees ◽  
Insertion Process

Simple families of increasing trees can be constructed from simply generated tree families, if one considers for every tree of size $n$ all its increasing labellings, i.$\,$e. labellings of the nodes by distinct integers of the set $\{1, \dots, n\}$ in such a way that each sequence of labels along any branch starting at the root is increasing. Three such tree families are of particular interest: recursive trees, plane-oriented recursive trees and binary increasing trees. We study the quantity number of descendants of node $j$ in a random tree of size $n$ and give closed formulæ for the probability distribution and all factorial moments for those subclass of tree families, which can be constructed via an insertion process. Furthermore limiting distribution results of this parameter are given.

scholarly journals Label-based parameters in increasing trees

2006 ◽  
Vol DMTCS Proceedings vol. AG,... (Proceedings) ◽  
Author(s):  
Markus Kuba ◽  
Alois Panholzer
Keyword(s):  
Probability Distribution ◽  
Random Variable ◽  
Growth Behavior ◽  
Limiting Distribution ◽  
Factorial Moments ◽  
International Audience ◽  
Recursive Trees ◽  
Insertion Process

International audience Grown simple families of increasing trees are a subclass of increasing trees, which can be constructed by an insertion process. Three such tree families contained in the grown simple families of increasing trees are of particular interest: $\textit{recursive trees}$, $\textit{plane-oriented recursive trees}$ and $\textit{binary increasing trees}$. Here we present a general approach for the analysis of a number of label-based parameters in a random grown simple increasing tree of size $n$ as, e.g., $\textit{the degree of the node labeled j}$, $\textit{the subtree-size of the node labeled j}$, etc. Further we apply the approach to the random variable $X_{n,j,a}$, which counts the number of size-$a$ branches attached to the node labeled $j$ (= subtrees of size $a$ rooted at the children of the node labeled $j$) in a random grown simple increasing tree of size $n$. We can give closed formulæ for the probability distribution and the factorial moments. Furthermore limiting distribution results for $X_{n,j,a}$ are given dependent on the growth behavior of $j=j(n)$ compared to $n$.

scholarly journals ON WEIGHTED PATH LENGTHS AND DISTANCES IN INCREASING TREES

2007 ◽  
Vol 21 (3) ◽  
pp. 419-433 ◽  
Author(s):  
M. Kuba ◽  
A. Panholzer
Keyword(s):  
Divisible Distribution ◽  
Limiting Distribution ◽  
Infinitely Divisible Distribution ◽  
Infinitely Divisible ◽  
Path Lengths ◽  
Recursive Trees ◽  
Insertion Process

We study weighted path lengths (depths) and distances for increasing tree families. For those subclasses of increasing tree families, which can be constructed via an insertion process (e.g., recursive trees, plane-oriented recursive trees, and binary increasing trees), we can determine the limiting distribution that can be characterized as a generalized Dickman's infinitely divisible distribution.

scholarly journals Labels distance in bucket recursive trees with variable capacities of buckets

2021 ◽  
Vol 13 (2) ◽  
pp. 413-426
Author(s):  
S. Naderi ◽  
R. Kazemi ◽  
M. H. Behzadi
Keyword(s):  
Partial Differential Equations ◽  
Probability Distribution ◽  
Differential Equations ◽  
Generating Function ◽  
Generating Functions ◽  
Function Approach ◽  
Closed Formula ◽  
Tree Model ◽  
Partial Differential ◽  
Recursive Trees

Abstract The bucket recursive tree is a natural multivariate structure. In this paper, we apply a trivariate generating function approach for studying of the depth and distance quantities in this tree model with variable bucket capacities and give a closed formula for the probability distribution, the expectation and the variance. We show as j → ∞, lim-iting distributions are Gaussian. The results are obtained by presenting partial differential equations for moment generating functions and solving them.

scholarly journals Corners in Tree-Like Tableaux

10.37236/5712 ◽  
2016 ◽  
Vol 23 (4) ◽  
Author(s):  
Paweł Hitczenko ◽  
Amanda Lohss
Keyword(s):  
Exclusion Process ◽  
Limiting Distribution ◽  
Tree Structure ◽  
Random Tree ◽  
Type B ◽  
Combinatorial Objects ◽  
Simple Exclusion Process ◽  
Asymmetric Simple Exclusion ◽  
Simple Exclusion ◽  
Symmetric Tree

In this paper, we study tree–like tableaux, combinatorial objects which exhibit a natural tree structure and are connected to the partially asymmetric simple exclusion process (PASEP). There was a conjecture made on the total number of corners in tree–like tableaux and the total number of corners in symmetric tree–like tableaux. In this paper, we prove both conjectures. Our proofs are based off of the bijection with permutation tableaux or type–B permutation tableaux and consequently, we also prove results for these tableaux. In addition, we derive the limiting distribution of the number of occupied corners in random tree–like tableaux and random symmetric tree–like tableaux.

The distribution and moments of the number of components of a random function

1990 ◽  
Vol 27 (01) ◽  
pp. 202-207 ◽  
Author(s):  
Joseph Kupka
Keyword(s):  
Probability Distribution ◽  
Random Function ◽  
Simple Formula ◽  
Computer Calculation ◽  
Factorial Moments ◽  
Number Of Components ◽  
Mean And Variance ◽  
The Mean

A relatively simple formula is presented for the probability distribution of the number K of components of a random function. This formula facilitates the (computer) calculation of the factorial moments of K and yields new expressions for the mean and variance of K.

Regenerative stochastic processes

1955 ◽  
Vol 232 (1188) ◽  
pp. 6-31 ◽  
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Probability Distribution ◽  
Stochastic Processes ◽  
Wide Class ◽  
Initial Conditions ◽  
Limiting Distribution ◽  
Regenerative Processes ◽  
Exponential Distributions ◽  
Negative Exponential

A wide class of stochastic processes, called regenerative, is defined, and it is shown that under general conditions the instantaneous probability distribution of such a process tends with time to a unique limiting distribution, whatever the initial conditions. The general results are then applied to 'S.M.-processes’, a generalization of Markov chains, and it is shown that the limiting distribution of the process may always be obtained by assuming negative exponential distributions for the ‘waits’ in the different ‘states’. Lastly, the behaviour of integrals of regenerative processes is considered and, amongst other results, an ergodic and a multi-dimensional central limit theorem are proved.

Asymptotics of the Hurwitz Binomial Distribution Related to Mixed Poisson Galton–Watson Trees

2001 ◽  
Vol 10 (3) ◽  
pp. 203-211 ◽  
Author(s):  
JÜRGEN BENNIES ◽  
JIM PITMAN
Keyword(s):  
Probability Distribution ◽  
Asymptotic Behaviour ◽  
Binomial Distribution ◽  
Random Tree ◽  
Binomial Theorem ◽  
Offspring Distribution

Hurwitz's extension of Abel's binomial theorem defines a probability distribution on the set of integers from 0 to n. This is the distribution of the number of non-root vertices of a fringe subtree of a suitably defined random tree with n + 2 vertices. The asymptotic behaviour of this distribution is described in a limiting regime in which the fringe subtree converges in distribution to a Galton–Watson tree with a mixed Poisson offspring distribution.

Asymptotic results on Hoppe trees and their variations

2020 ◽  
Vol 57 (2) ◽  
pp. 441-457
Author(s):  
Ella Hiesmayr ◽  
Ümit Işlak
Keyword(s):  
Rooted Tree ◽  
Random Tree ◽  
Equal Probability ◽  
Asymptotic Results ◽  
Tree Model ◽  
Recursive Tree ◽  
Weighted Trees ◽  
Number Of Leaves ◽  
Recursive Trees ◽  
Number Of Branches

AbstractA uniform recursive tree on n vertices is a random tree where each possible $(n-1)!$ labelled recursive rooted tree is selected with equal probability. We introduce and study weighted trees, a non-uniform recursive tree model departing from the recently introduced Hoppe trees. This class generalizes both uniform recursive trees and Hoppe trees, providing diversity among the nodes and making the model more flexible for applications. We analyse the number of leaves, the height, the depth, the number of branches, and the size of the largest branch in these weighted trees.

scholarly journals On the spectral dimension of random trees

2006 ◽  
Vol DMTCS Proceedings vol. AG,... (Proceedings) ◽  
Author(s):  
Bergfinnur Durhuus ◽  
Thordur Jonsson ◽  
John Wheater
Keyword(s):  
Probability Distribution ◽  
Random Trees ◽  
Random Tree ◽  
Spectral Dimension ◽  
Fixed Size ◽  
Infinite Trees ◽  
Linear Chains ◽  
International Audience

International audience We determine the spectral dimensions of a variety of ensembles of infinite trees. Common to the ensembles considered is that sample trees have a distinguished infinite spine at whose vertices branches can be attached according to some probability distribution. In particular, we consider a family of ensembles of $\textit{combs}$, whose branches are linear chains, with spectral dimensions varying continuously between $1$ and $3/2$. We also introduce a class of ensembles of infinite trees, called $\textit{generic random trees}$, which are obtained as limits of ensembles of finite trees conditioned to have fixed size $N$, as $N \to \infty$. Among these ensembles is the so-called uniform random tree. We show that generic random trees have spectral dimension $d_s=4/3$.

scholarly journals Structure of spanning trees on the two-dimensional Sierpinski gasket

2011 ◽  
Vol Vol. 12 no. 3 (Combinatorics) ◽  
Author(s):  
Shu-Chiuan Chang ◽  
Lung-Chi Chen
Keyword(s):  
Probability Distribution ◽  
Spanning Tree ◽  
Spanning Trees ◽  
Sierpinski Gasket ◽  
Rational Numbers ◽  
Limiting Distribution ◽  
Two Dimensional ◽  
Sierpiński Gasket ◽  
Average Probability ◽  
International Audience

Combinatorics International audience Consider spanning trees on the two-dimensional Sierpinski gasket SG(n) where stage n is a non-negative integer. For any given vertex x of SG(n), we derive rigorously the probability distribution of the degree j ∈{1,2,3,4} at the vertex and its value in the infinite n limit. Adding up such probabilities of all the vertices divided by the number of vertices, we obtain the average probability distribution of the degree j. The corresponding limiting distribution φj gives the average probability that a vertex is connected by 1, 2, 3 or 4 bond(s) among all the spanning tree configurations. They are rational numbers given as φ1=10957/40464, φ2=6626035/13636368, φ3=2943139/13636368, φ4=124895/4545456.

Sign in / Sign up

Export Citation Format

Share Document