On the distribution of distances in recursive trees

1996 ◽  
Vol 33 (03) ◽  
pp. 749-757 ◽  
Author(s):  
Robert P. Dobrow
Keyword(s):  
Exact Distribution ◽  
Recursive Tree ◽  
Asymptotically Normal ◽  
Ancient Manuscripts ◽  
The Law ◽  
Recursive Trees ◽  
Pyramid Schemes ◽  
Bernoulli Distributions ◽  
Family Trees

Recursive trees have been used to model such things as the spread of epidemics, family trees of ancient manuscripts, and pyramid schemes. A tree Tn with n labeled nodes is a recursive tree if n = 1, or n > 1 and Tn can be constructed by joining node n to a node of some recursive tree Tn– 1. For arbitrary nodes i < n in a random recursive tree we give the exact distribution of Xi,n , the distance between nodes i and n. We characterize this distribution as the convolution of the law of Xi,j+ 1 and n – i – 1 Bernoulli distributions. We further characterize the law of Xi,j+ 1 as a mixture of sums of Bernoullis. For i = in growing as a function of n, we show that is asymptotically normal in several settings.

On the distribution of distances in recursive trees

1996 ◽  
Vol 33 (3) ◽  
pp. 749-757 ◽  
Author(s):  
Robert P. Dobrow
Keyword(s):  
Exact Distribution ◽  
Recursive Tree ◽  
Asymptotically Normal ◽  
Ancient Manuscripts ◽  
The Law ◽  
Recursive Trees ◽  
Pyramid Schemes ◽  
Bernoulli Distributions ◽  
Family Trees

Recursive trees have been used to model such things as the spread of epidemics, family trees of ancient manuscripts, and pyramid schemes. A tree Tn with n labeled nodes is a recursive tree if n = 1, or n > 1 and Tn can be constructed by joining node n to a node of some recursive tree Tn–1. For arbitrary nodes i < n in a random recursive tree we give the exact distribution of Xi,n, the distance between nodes i and n. We characterize this distribution as the convolution of the law of Xi,j+1 and n – i – 1 Bernoulli distributions. We further characterize the law of Xi,j+1 as a mixture of sums of Bernoullis. For i = in growing as a function of n, we show that is asymptotically normal in several settings.

On the size of paged recursive trees

2017 ◽  
Vol 09 (02) ◽  
pp. 1750021
Author(s):  
Mehri Javanian
Keyword(s):  
Contraction Method ◽  
Recursive Tree ◽  
Asymptotically Normal ◽  
Recursive Trees ◽  
Normal Formula

A paged recursive tree is constructed as a recursive tree except that it depends on an integer parameter [Formula: see text] representing a page capacity, small subtrees with size [Formula: see text]. We investigate the number of nodes [Formula: see text] (size of the tree) in paged recursive trees built from labels [Formula: see text]. The expectation and variance of [Formula: see text] are derived, and it is also shown that [Formula: see text] is asymptotically normal. [Formula: see text] as [Formula: see text] by applying the contraction method.

scholarly journals A Strong Law for the Size of Yule M-Oriented Recursive Trees

2012 ◽  
Vol 8 (2) ◽  
pp. 67-72
Author(s):  
Mehri Javanian ◽  
Mohammad Q. Vahidi-Asl
Keyword(s):  
Exact Distribution ◽  
Birth Rates ◽  
Strong Law ◽  
Recursive Tree ◽  
Yule Process ◽  
Recursive Trees

Abstract Let Nt be the total number of nodes in a Yule m-oriented recursive tree at time t. Then {Nt : t ∈ [0;1)} is a Yule process with birth rates λn = (m(n - 1) + 1)λ for n ≥ 1, where N0 = 1. In this paper, we first give the exact distribution of Nt, then prove that , almost surely

Limiting Distributions for Path Lengths in Recursive Trees

1991 ◽  
Vol 5 (1) ◽  
pp. 53-59 ◽  
Author(s):  
Hosam M. Mahmoud
Keyword(s):  
Path Length ◽  
Elementary Proof ◽  
Random Variable ◽  
Limiting Distribution ◽  
Limiting Distributions ◽  
Quadratic Mean ◽  
Recursive Tree ◽  
Asymptotically Normal ◽  
Path Lengths ◽  
Recursive Trees

The depth of insertion and the internal path length of recursive trees are studied. Luc Devroye has recently shown that the depth of insertion in recursive trees is asymptotically normal. We give a direct alternative elementary proof of this fact. Furthermore, via the theory of martingales, we show that In, the internal path length of a recursive tree of order n, converges to a limiting distribution. In fact, we show that there exists a random variable I such that (In – n In n)/n→I almost surely and in quadratic mean, as n → α. The method admits, in passing, the calculation of the first two moments of In.

scholarly journals On the Zagreb Index of Random Recursive Trees

2011 ◽  
Vol 48 (04) ◽  
pp. 1189-1196 ◽  
Author(s):  
Qunqiang Feng ◽  
Zhishui Hu
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Asymptotic Normality ◽  
Random Process ◽  
Central Limit ◽  
Topological Indices ◽  
Zagreb Index ◽  
Martingale Central Limit Theorem ◽  
Recursive Tree ◽  
Recursive Trees

We investigate the Zagreb index, one of the topological indices, of random recursive trees in this paper. Through a recurrence equation, the first two moments ofZn, the Zagreb index of a random recursive tree of sizen, are obtained. We also show that the random process {Zn− E[Zn],n≥ 1} is a martingale. Then the asymptotic normality of the Zagreb index of a random recursive tree is given by an application of the martingale central limit theorem. Finally, two other topological indices are also discussed in passing.

Phase Changes in the Topological Indices of Scale-Free Trees

2013 ◽  
Vol 50 (02) ◽  
pp. 516-532 ◽  
Author(s):  
Qunqiang Feng ◽  
Zhishui Hu
Keyword(s):  
Nonnegative Integer ◽  
Topological Indices ◽  
Critical Values ◽  
Phase Changes ◽  
Zagreb Index ◽  
Asymptotic Behaviors ◽  
Scale Free ◽  
Recursive Tree ◽  
Free Tree ◽  
Recursive Trees

A scale-free tree with the parameter β is very close to a star if β is just a bit larger than −1, whereas it is close to a random recursive tree if β is very large. Through the Zagreb index, we consider the whole scene of the evolution of the scale-free trees model as β goes from −1 to + ∞. The critical values of β are shown to be the first several nonnegative integer numbers. We get the first two moments and the asymptotic behaviors of this index of a scale-free tree for all β. The generalized plane-oriented recursive trees model is also mentioned in passing, as well as the Gordon-Scantlebury and the Platt indices, which are closely related to the Zagreb index.

scholarly journals Deterministic Edge Weights in Increasing Tree Families

2009 ◽  
Vol 19 (1) ◽  
pp. 99-119
Author(s):  
MARKUS KUBA ◽  
STEPHAN WAGNER
Keyword(s):  
Normal Limit ◽  
Random Trees ◽  
Edge Weight ◽  
Combinatorial Approach ◽  
Related Parameter ◽  
Scale Free ◽  
Recursive Tree ◽  
Combinatorial Model ◽  
Recursive Trees

In this work we study edge weights for two specific families of increasing trees, which include binary increasing trees and plane-oriented recursive trees as special instances, where plane-oriented recursive trees serve as a combinatorial model of scale-free random trees given by the m = 1 case of the Barabási–Albert model. An edge e = (k, l), connecting the nodes labelled k and l, respectively, in an increasing tree, is associated with the weight we = |k − l|. We are interested in the distribution of the number of edges with a fixed edge weight j in a random generalized plane-oriented recursive tree or random d-ary increasing tree. We provide exact formulas for expectation and variance and prove a normal limit law for this quantity. A combinatorial approach is also presented and applied to a related parameter, the maximum edge weight.

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 Learning to Embed Sentences Using Attentive Recursive Trees

2019 ◽  
Vol 33 ◽  
pp. 6991-6998
Author(s):  
Jiaxin Shi ◽  
Lei Hou ◽  
Juanzi Li ◽  
Zhiyuan Liu ◽  
Hanwang Zhang
Keyword(s):  
Deep Learning ◽  
State Of The Art ◽  
The State ◽  
Feature Representation ◽  
Tree Structure ◽  
Tree Model ◽  
Training Strategy ◽  
Recursive Tree ◽  
Recursive Trees ◽  
Embedding Methods

Sentence embedding is an effective feature representation for most deep learning-based NLP tasks. One prevailing line of methods is using recursive latent tree-structured networks to embed sentences with task-specific structures. However, existing models have no explicit mechanism to emphasize taskinformative words in the tree structure. To this end, we propose an Attentive Recursive Tree model (AR-Tree), where the words are dynamically located according to their importance in the task. Specifically, we construct the latent tree for a sentence in a proposed important-first strategy, and place more attentive words nearer to the root; thus, AR-Tree can inherently emphasize important words during the bottomup composition of the sentence embedding. We propose an end-to-end reinforced training strategy for AR-Tree, which is demonstrated to consistently outperform, or be at least comparable to, the state-of-the-art sentence embedding methods on three sentence understanding tasks.

scholarly journals Random Recursive Trees and Preferential Attachment Trees are Random Split Trees

2018 ◽  
Vol 28 (1) ◽  
pp. 81-99 ◽  
Author(s):  
SVANTE JANSON
Keyword(s):  
Preferential Attachment ◽  
Recursive Tree ◽  
The Common ◽  
Split Trees ◽  
Recursive Trees

We consider linear preferential attachment trees, and show that they can be regarded as random split trees in the sense of Devroye (1999), although with infinite potential branching. In particular, this applies to the random recursive tree and the standard preferential attachment tree. An application is given to the sum over all pairs of nodes of the common number of ancestors.

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