Profiles of random trees: Plane‐oriented recursive trees

Hwang, Hsien‐Kuei

doi:10.1002/rsa.20139

Cited by 32 publications

(39 citation statements)

References 57 publications

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“…for all 1 ≤ s ≤ m. Accordingly, we get E[W 2 n−1,s | F n−1,0 ], as stated in the proposition. On the other hand, we take the expectation of both sides of (9) and obtain:…”

Section: Nodes Of Small Outdegreesmentioning

confidence: 99%

On Nodes of Small Degrees and Degree Profile in Preferential Dynamic Attachment Circuits

Zhang

Mahmoud

2019

Methodol Comput Appl Probab

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We investigate the joint distribution of nodes of small degrees and the degree profile in preferential dynamic attachment circuits. In particular, we study the joint asymptotic distribution of the number of the nodes of outdegree 0 (terminal nodes) and outdegree 1 in a very large circuit. The expectation and variance of the number of those two types of nodes are both asymptotically linear with respect to the age of the circuit. We show that the numbers of nodes of outdegree 0 and 11 asymptotically follow a twodimensional Gaussian law via multivariate martingale methods. We also study the exact distribution of the degree of a node, as the circuit ages, via a series of Pólya-Eggenberger urn models with "hiccups" in between. The exact expectation and variance of the degree of nodes are determined by recurrence methods. Phase transitions of these degrees are discussed briefly. This is an extension of the abstract [25].

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“…for all 1 ≤ s ≤ m. Accordingly, we get E[W 2 n−1,s | F n−1,0 ], as stated in the proposition. On the other hand, we take the expectation of both sides of (9) and obtain:…”

Section: Nodes Of Small Outdegreesmentioning

confidence: 99%

On Nodes of Small Degrees and Degree Profile in Preferential Dynamic Attachment Circuits

Zhang

Mahmoud

2019

Methodol Comput Appl Probab

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“…This is nothing but the preferential attachment model of Barabasi and Albert (see Albert-Barabasi-Jeong [2], or Albert-Barabasi [1]), which for a single parent is a special case of the linear recursive trees or plane-oriented recursive tree. For this model, the parameter R n was studied by Mahmoud [24], and the height by Pittel [30] and Biggins-Grey [4], and in a rather general setting by BroutinDevroye [5]: the height is in probability (1.7956...+o(1)) log n. The profile (number of nodes at each depth level) was studied by Hwang [21], [22] and Sulzbach [36].…”

Section: Bibliographic Remarks and Possible Extensionsmentioning

confidence: 99%

Long and short paths in uniform random recursive dags

Devroye

Janson

2011

Ark. Mat.

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In a uniform random recursive k-dag, there is a root, 0, and each node in turn, from 1 to n, chooses k uniform random parents from among the nodes of smaller index. If S_n is the shortest path distance from node n to the root, then we determine the constant \sigma such that S_n/log(n) tends to \sigma in probability as n tends to infinity. We also show that max_{1 \le i \le n} S_i/log(n) tends to \sigma in probability.Comment: 16 page

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“…The moment sequence (41) is easily checked to have the property of uniquely characterizing the distribution; see Hwang (2005) for similar details.…”

Section: Asymptotics Of Pmentioning

confidence: 99%

Profiles of Random Trees: Limit Theorems for Random Recursive Trees and Binary Search Trees

2006

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We prove convergence in distribution for the profile (the number of nodes at each level), normalized by its mean, of random recursive trees when the limit ratio˛of the level and the logarithm of tree size lies in OE0; e/. Convergence of all moments is shown to hold only for˛2 OE0; 1 (with only convergence of finite moments when˛2 .1; e/). When the limit ratio is 0 or 1 for which the limit laws are both constant, we prove asymptotic normality for˛D 0 and a "quicksort type" limit law for˛D 1, the latter case having additionally a small range where there is no fixed limit law. Our tools are based on contraction method and method of moments. Similar phenomena also hold for other classes of trees; we apply our tools to binary search trees and give a complete characterization of the profile. The profiles of these random trees represent concrete examples for which the range of convergence in distribution differs from that of convergence of all moments.

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Profiles of random trees: Plane‐oriented recursive trees

Cited by 32 publications

References 57 publications

On Nodes of Small Degrees and Degree Profile in Preferential Dynamic Attachment Circuits

On Nodes of Small Degrees and Degree Profile in Preferential Dynamic Attachment Circuits

Long and short paths in uniform random recursive dags

Profiles of Random Trees: Limit Theorems for Random Recursive Trees and Binary Search Trees

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