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 Profiles of random trees: correlation and width of random recursive trees and binary search trees

2005 ◽  
Vol 37 (02) ◽  
pp. 321-341 ◽  
Author(s):  
Michael Drmota ◽  
Hsien-Kuei Hwang
Keyword(s):  
Asymptotic Estimate ◽  
Correlation Coefficients ◽  
Singularity Analysis ◽  
Binary Search ◽  
Random Trees ◽  
The Other ◽  
Binary Search Trees ◽  
Search Trees ◽  
Expected Width ◽  
Recursive Trees

In a tree, a level consists of all those nodes that are the same distance from the root. We derive asymptotic approximations to the correlation coefficients of two level sizes in random recursive trees and binary search trees. These coefficients undergo sharp sign-changes when one level is fixed and the other is varying. We also propose a new means of deriving an asymptotic estimate for the expected width, which is the number of nodes at the most abundant level. Crucial to our methods of proof is the uniformity achieved by singularity analysis.

scholarly journals Profiles of random trees: correlation and width of random recursive trees and binary search trees

2005 ◽  
Vol 37 (2) ◽  
pp. 321-341 ◽  
Author(s):  
Michael Drmota ◽  
Hsien-Kuei Hwang
Keyword(s):  
Asymptotic Estimate ◽  
Correlation Coefficients ◽  
Singularity Analysis ◽  
Binary Search ◽  
Random Trees ◽  
The Other ◽  
Binary Search Trees ◽  
Search Trees ◽  
Expected Width ◽  
Recursive Trees

In a tree, a level consists of all those nodes that are the same distance from the root. We derive asymptotic approximations to the correlation coefficients of two level sizes in random recursive trees and binary search trees. These coefficients undergo sharp sign-changes when one level is fixed and the other is varying. We also propose a new means of deriving an asymptotic estimate for the expected width, which is the number of nodes at the most abundant level. Crucial to our methods of proof is the uniformity achieved by singularity analysis.

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 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 The b-chromatic number of power graphs

2003 ◽  
Vol Vol. 6 no. 1 ◽  
Author(s):  
Brice Effantin ◽  
Hamamache Kheddouci
Keyword(s):  
Chromatic Number ◽  
Binary Trees ◽  
Proper Coloring ◽  
Power Cycles ◽  
International Audience

International audience The b-chromatic number of a graph G is defined as the maximum number k of colors that can be used to color the vertices of G, such that we obtain a proper coloring and each color i, with 1 ≤ i≤ k, has at least one representant x_i adjacent to a vertex of every color j, 1 ≤ j ≠ i ≤ k. In this paper, we discuss the b-chromatic number of some power graphs. We give the exact value of the b-chromatic number of power paths and power complete binary trees, and we bound the b-chromatic number of power cycles.

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 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 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.

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