Random Trees

An Interplay Between Combinatorics and Probability

By: Michael Drmota

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Hardcover | 22 December 2008

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478 Pages

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23.5 x 16.51 x 3.81

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Trees are a fundamental object in graph theory and combinatorics as well as a basic object for data structures and algorithms in computer science. During thelastyearsresearchrelatedto(random)treeshasbeenconstantlyincreasing and several asymptotic and probabilistic techniques have been developed in order to describe characteristics of interest of large trees in di?erent settings. Thepurposeofthisbookistoprovideathoroughintroductionintovarious aspects of trees in randomsettings anda systematic treatment ofthe involved mathematicaltechniques. It shouldserveasa referencebookaswellasa basis for future research. One major conceptual aspect is to connect combinatorial and probabilistic methods that range from counting techniques (generating functions, bijections) over asymptotic methods (singularity analysis, saddle point techniques) to various sophisticated techniques in asymptotic probab- ity (convergence of stochastic processes, martingales). However, the reading of the book requires just basic knowledge in combinatorics, complex analysis, functional analysis and probability theory of master degree level. It is also part of concept of the book to provide full proofs of the major results even if they are technically involved and lengthy.

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You Can Find This Book In

Non-Fiction Mathematics Applied Mathematics Stochastics Nature & The Natural World Trees, Mushrooms, Wildflowers & Plants Combinatorics & Graph Theory Algebra Probability & Statistics Discrete Mathematics Number Theory Computing & I.T.

Computer Programming & Software Development Algorithms & Data Structures Graphical & Digital Media Applications 3D Graphics & Modelling Databases Computer Science Artificial Intelligence

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Paperback

Published: 13th October 2010

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Classes of Random Trees	p. 1
Basic Notions	p. 2
Rooted Versus Unrooted trees	p. 2
Plane Versus Non-Plane trees	p. 3
Labelled Versus Unlabelled Trees	p. 3
Combinatorial Trees	p. 4
Binary Trees	p. 5
Planted Plane Trees	p. 6
Labelled Trees	p. 7
Labelled Plane Trees	p. 8
Unlabelled Trees	p. 8
Unlabelled Plane Trees	p. 9
Simply Generated Trees - Galton-Watson Trees	p. 9
Recursive Trees	p. 13
Non-Plane Recursive Trees	p. 13
Plane Oriented Recursive Trees	p. 14
Increasing Trees	p. 15
Search Trees	p. 17
Binary Search Trees	p. 18
Fringe Balanced m-Ary Search Trees	p. 19
Digital Search Trees	p. 21
Tries	p. 22
Generating Functions	p. 25
Counting with Generating Functions	p. 26
Generating Functions and Combinatorial Constructions	p. 27
Pólya's Theory of Counting	p. 33
Lagrange Inversion Formula	p. 36
Asymptotics with Generating Functions	p. 37
Asymptotic Transfers	p. 38
Functional Equations	p. 43
Asymptotic Normality and Functional Equations	p. 46
Transfer of Singularities	p. 54
Systems of Functional Equations	p. 62
Advanced Tree Counting	p. 69
Generating Functions and Combinatorial Trees	p. 70
Binary and m-ary Trees	p. 70
Planted Plane Trees	p. 71
Labelled Trees	p. 73
Simply Generated Trees - Galton-Watson Trees	p. 75
Unrooted Trees	p. 77
Trees Embedded in the Plane	p. 81
Additive Parameters in Trees	p. 82
Simply Generated Trees - Galton-Watson Trees	p. 84
Unrooted Trees	p. 87
Patterns in Trees	p. 90
Planted, Rooted and Unrooted Trees	p. 91
Generating Functions for Planted Rooted Trees	p. 92
Rooted and Unrooted Trees	p. 99
Asymptotic Behaviour	p. 101
The Shape of Galton-Watson Trees and Pólya Trees	p. 107
The Continuum Random Tree	p. 108
Depth-First Search of a Rooted Tree	p. 108
Real Trees	p. 109
Galton-Watson Trees and the Continuum Random Tree	p. 111
The Profile of Galton-Watson Trees	p. 115
The Distribution of the Local Time	p. 118
Weak Convergence of Continuous Stochastic Processes	p. 120
Combinatorics on the Profile of Galton-Watson Trees	p. 125
Asymptotic Analysis of the Main Recurrence	p. 126
Finite Dimensional Limiting Distributions	p. 129
Tightness	p. 134
The Height of Galton-Watson Trees	p. 139
Depth-First Search	p. 149
The Profile of Pólya Trees	p. 154
Combinatorial Setup	p. 154
Asymptotic Analysis of the Main Recurrence	p. 156
Finite Dimensional Limiting Distributions	p. 164
Tightness	p. 168
The Height of Pólya Trees	p. 177
The Vertical Profile of Trees	p. 187
Quadrangulations and Embedded Trees	p. 188
Profiles of Trees and Random Measures	p. 196
General Profiles	p. 196
Space Embedded Trees and ISE	p. 196
The Distribution of the ISE	p. 204
Combinatorics on Embedded Trees	p. 207
Embedded Trees with Increments ±1	p. 207
Embedded Trees with Increments 0, ±1	p. 214
Naturally Embedded Binary Trees	p. 216
Asymptotics on Embedded Trees	p. 219
Trees with Small Labels	p. 219
The Number of Nodes of Given Label	p. 225
The Number of Nodes of Large Labels	p. 229
Embedded Trees with Increments 0 and ±1	p. 235
Naturally Embedded Binary Trees	p. 235
Recursive Trees and Binary Search Trees	p. 237
Permutations and Trees	p. 238
Permutations and Recursive Trees	p. 239
Permutations and Binary Search Trees	p. 246
Generating Functions and Basic Statistics	p. 247
Generating Functions for Recursive Trees	p. 248
Generating Functions for Binary Search Trees	p. 249
Generating Functions for Plane Oriented Recursive Trees	p. 251
The Degree Distribution of Recursive Trees	p. 253
The Insertion Depth	p. 262
The Profile of Recursive Trees	p. 265
The Martingale Method	p. 266
The Moment Method	p. 275
The Contraction Method	p. 278
The Height of Recursive Trees	p. 280
Profile and Height of Binary Search Trees and Related Trees	p. 291
The Profile of Binary Search Trees and Related Trees	p. 291
The Height of Binary Search Trees and Related Trees	p. 300
Tries and Digital Search Trees	p. 307
The Profile of Tries	p. 308
Generating Functions for the Profile	p. 308
The Expected Profile of Tries	p. 311
The Limiting Distribution of the Profile of Tries	p. 321
The Height of Tries	p. 323
Symmetric Tries	p. 324
The Profile of Digital Search Trees	p. 325
Generating Functions for the Profile	p. 325
The Expected Profile of Digital Search Trees	p. 327
Symmetric Digital Search Trees	p. 337
Recursive Algorithms and the Contraction Method	p. 343
The Number of Comparisons in Quicksort	p. 345
The L2 Setting of the Contraction Method	p. 350
A General Type of Recurrence	p. 350
A General L2 Convergence Theorem	p. 352
Applications of the L2 Setting	p. 357
Limitations of the L2 Setting and Extensions	p. 361
The Zolotarev Metric	p. 362
Degenerate Limit Equations	p. 363
Planar Graphs	p. 365
Basic Notions	p. 366
Counting Planar Graphs	p. 368
Outerplanar Graphs	p. 368
Series-Parallel Graphs	p. 376
Quadrangulations and Planar Maps	p. 382
Planar Graphs	p. 389
Outerplanar Graphs	p. 396
The Degree Distribution of Outerplanar Graphs	p. 396
Vertices of Given Degree in Dissections	p. 400
Vertices of Given Degree in 2-Connected Outerplanar Graphs	p. 404
Vertices of Given Degree in Connected Outerplanar Graphs	p. 406
Series-Parallel Graphs	p. 408
The Degree Distribution of Series-Parallel Graphs	p. 408
Vertices of Given Degree in Series-Parallel Networks	p. 415
Vertices of Given Degree in 2-Connected Series-Parallel Graphs	p. 416
Vertices of Given Degree in Connected Series-Parallel Graphs	p. 419
All Planar Graphs	p. 420
The Degree of a Rooted Vertex	p. 421
Singular Expansions	p. 425
Degree Distribution for Planar Graphs	p. 429
Vertices of Degree 1 or 2 in Planar Graphs	p. 433
Appendix	p. 439
References	p. 445
Index	p. 455
Table of Contents provided by Ingram. All Rights Reserved.

Random Trees

An Interplay Between Combinatorics and Probability

At a Glance

Hardcover

Shipping

How to return your order

Defective items

You Can Find This Book In

Other Editions and Formats

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