scholarly journals Isolating nodes in recursive trees

2008 ◽  
Vol 76 (3) ◽  
pp. 258-280 ◽  
Author(s):  
Markus Kuba ◽  
Alois Panholzer
Keyword(s):  
Recursive Trees

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 On the Distribution of Depths in Increasing Trees

10.37236/409 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Markus Kuba ◽  
Stephan Wagner
Keyword(s):  
Common Ancestor ◽  
Short Note ◽  
Bijective Proof ◽  
Lowest Common Ancestor ◽  
Recursive Trees

By a theorem of Dobrow and Smythe, the depth of the $k$th node in very simple families of increasing trees (which includes, among others, binary increasing trees, recursive trees and plane ordered recursive trees) follows the same distribution as the number of edges of the form $j-(j+1)$ with $j < k$. In this short note, we present a simple bijective proof of this fact, which also shows that the result actually holds within a wider class of increasing trees. We also discuss some related results that follow from the bijection as well as a possible generalization. Finally, we use another similar bijection to determine the distribution of the depth of the lowest common ancestor of two nodes.

ON SEVERAL PROPERTIES OF A CLASS OF PREFERENTIAL ATTACHMENT TREES—PLANE-ORIENTED RECURSIVE TREES

2020 ◽  
pp. 1-19
Author(s):  
Panpan Zhang
Keyword(s):  
Continuous Time ◽  
Preferential Attachment ◽  
Mass Function ◽  
Probability Mass Function ◽  
Zagreb Index ◽  
Exact Probability ◽  
Probability Mass ◽  
Proper Scaling ◽  
Recursive Trees ◽  
Degree Variable

In this paper, several properties of a class of trees presenting preferential attachment phenomenon—plane-oriented recursive trees (PORTs) are uncovered. Specifically, we investigate the degree profile of a PORT by determining the exact probability mass function of the degree of a node with a fixed label. We compute the expectation and the variance of degree variable via a Pólya urn approach. In addition, we study a topological index, Zagreb index, of this class of trees. We calculate the exact first two moments of the Zagreb index (of PORTs) by using recurrence methods. Lastly, we determine the limiting degree distribution in PORTs that grow in continuous time, where the embedding is done in a Poissonization framework. We show that it is exponential after proper scaling.

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 Structural and spectral properties of a family of deterministic recursive trees: rigorous solutions

2009 ◽  
Vol 42 (16) ◽  
pp. 165103 ◽  
Author(s):  
Yi Qi ◽  
Zhongzhi Zhang ◽  
Bailu Ding ◽  
Shuigeng Zhou ◽  
Jihong Guan
Keyword(s):  
Spectral Properties ◽  
Recursive Trees
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