Random trees: an interplay between combinatorics and probability
Out of research related to (random) trees, several asymptotic and probabilistic techniques have been developed to describe characteristics of large trees in different settings. The aim here is to provide an introduction to various aspects of trees in random settings and a systematic treatment of the...
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Wien ; New York
Springer
2009
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Summary: | Out of research related to (random) trees, several asymptotic and probabilistic techniques have been developed to describe characteristics of large trees in different settings. The aim here is to provide an introduction to various aspects of trees in random settings and a systematic treatment of the involved mathematical techniques. |
| Physical Description: | XVII, 458 Seiten Diagramme |
| ISBN: | 9783211753552 3211753559 |
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MARC
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| 245 | 1 | 0 | |a Random trees |b an interplay between combinatorics and probability |c by Michael Drmota |
| 264 | 1 | |a Wien ; New York |b Springer |c 2009 | |
| 300 | |a XVII, 458 Seiten |b Diagramme | ||
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| 520 | 3 | |a Out of research related to (random) trees, several asymptotic and probabilistic techniques have been developed to describe characteristics of large trees in different settings. The aim here is to provide an introduction to various aspects of trees in random settings and a systematic treatment of the involved mathematical techniques. | |
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Record in the Search Index
| _version_ | 1804138313995517952 |
|---|---|
| adam_text | Contents
Classes
of Random Trees
.................................. 1
1.1
Basic Notions
........................................... 2
1.1.1
Rooted Versus Unrooted trees
....................... 2
1.1.2
Plane Versus Non-Plane trees
....................... 3
1.1.3
Labelled Versus Unlabelled Trees
.................... 3
1.2
Combinatorial Trees
..................................... 4
1.2.1
Binary Trees
...................................... 5
1.2.2
Planted Plane Trees
............................... 6
1.2.3
Labelled Trees
.................................... 7
1.2.4
Labelled Plane Trees
............................... 8
1.2.5
Unlabelled Trees
.................................. 8
1.2.6
Unlabelled Plane Trees
............................. 9
1.2.7
Simply Generated Trees
-
Galton-Watson Trees
....... 9
1.3
Recursive Trees
......................................... 13
1.3.1
Non-Plane Recursive Trees
......................... 13
1.3.2
Plane Oriented Recursive Trees
..................... 14
1.3.3
Increasing Trees
................................... 15
1.4
Search Trees
............................................ 17
1.4.1
Binary Search Trees
............................... 18
1.4.2
Fringe Balanced m-Ary Search Trees
................. 19
1.4.3
Digital Search Trees
............................... 21
1.4.4
Tries
............................................. 22
Generating Functions
...................................... 25
2.1
Counting with Generating Functions
....................... 26
2.1.1
Generating Functions and Combinatorial Constructions
27
2.1.2
Pólya s
Theory of Counting
......................... 33
2.1.3 Lagrange
Inversion Formula
........................ 36
2.2
Asymptotics with Generating Functions
.................... 37
2.2.1
Asymptotic Transfers
.............................. 38
2.2.2
Functional Equations
.............................. 43
XIV Contents
2.2.3
Asymptotic Normality and Functional Equations
...... 46
2.2.4
Transfer of Singularities
............................ 54
2.2.5
Systems of Functional Equations
.................... 62
3
Advanced Tree Counting
.................................. 69
3.1
Generating Functions and Combinatorial Trees
.............. 70
3.1.1
Binary and rn-ary Trees
............................ 70
3.1.2
Planted Plane Trees
............................... 71
3.1.3
Labelled Trees
.................................... 73
3.1.4
Simply Generated Trees
-
Galton-Watson Trees
....... 75
3.1.5
Unrooted Trees
................................... 77
3.1.6
Trees Embedded in the Plane
....................... 81
3.2
Additive Parameters in Trees
............................. 82
3.2.1
Simply Generated Trees
-
Galton-Watson Trees
....... 84
3.2.2
Unrooted Trees
................................... 87
3.3
Patterns in Trees
........................................ 90
3.3.1
Planted, Rooted and Unrooted Trees
................. 91
3.3.2
Generating Functions for Planted Rooted Trees
....... 92
3.3.3
Rooted and Unrooted Trees
......................... 99
3.3.4
Asymptotic Behaviour
.............................101
4
The Shape of Galton-Watson Trees and
Pólya
Trees
.......107
4.1
The Continuum Random Tree
.............................108
4.1.1
Depth-First Search of a Rooted Tree
.................108
4.1.2
Real Trees
........................................109
4.1.3
Galton-Watson Trees and the Continuum Random Tree 111
4.2
The Profile of Galton-Watson Trees
........................115
4.2.1
The Distribution of the Local Time
..................118
4.2.2
Weak Convergence of Continuous Stochastic Processes
. 120
4.2.3
Combinatorics on the Profile of Galton-Watson Trees
.. 125
4.2.4
Asymptotic Analysis of the Main Recurrence
.........126
4.2.5
Finite Dimensional Limiting Distributions
............129
4.2.6
Tightness
........................................134
4.2.7
The Height of Galton-Watson Trees
..................139
4.2.8
Depth-First Search
................................149
4.3
The Profile of
Pólya
Trees
................................154
4.3.1
Combinatorial Setup
...............................154
4.3.2
Asymptotic Analysis of the Main Recurrence
.........156
4.3.3
Finite Dimensional Limiting Distributions
............164
4.3.4
Tightness
........................................168
4.3.5
The Height of
Pólya
Trees
..........................177
Contents
XV
The Vertical Profile of Trees
...............................187
5.1
Quadrangulations and Embedded Trees
....................188
5.2
Profiles of Trees and Random Measures
....................196
5.2.1
General Profiles
...................................196
5.2.2
Space Embedded Trees and ISE
.....................196
5.2.3
The Distribution of the ISE
.........................204
5.3
Combinatorics on Embedded Trees
........................207
5.3.1
Embedded Trees with Increments
±1 ................207
5.3.2
Embedded Trees with Increments
0, ±1 ..............214
5.3.3
Naturally Embedded Binary Trees
...................216
5.4
Asymptotics on Embedded Trees
..........................219
5.4.1
Trees with Small Labels
............................219
5.4.2
The Number of Nodes of Given Label
................225
5.4.3
The Number of Nodes of Large Labels
...............229
5.4.4
Embedded Trees with Increments
0
and
±1...........235
5.4.5
Naturally Embedded Binary Trees
...................235
Recursive Trees and Binary Search Trees
..................237
6.1
Permutations and Trees
..................................238
6.1.1
Permutations and Recursive Trees
...................239
6.1.2
Permutations and Binary Search Trees
...............246
6.2
Generating Functions and Basic Statistics
..................247
6.2.1
Generating Functions for Recursive Trees
.............248
6.2.2
Generating Functions for Binary Search Trees
.........249
6.2.3
Generating Functions for Plane Oriented Recursive
Trees
............................................251
6.2.4
The Degree Distribution of Recursive Trees
...........253
6.2.5
The Insertion Depth
...............................262
6.3
The Profile of Recursive Trees
.............................265
6.3.1
The Martingale Method
............................266
6.3.2
The Moment Method
..............................275
6.3.3
The Contraction Method
...........................278
6.4
The Height of Recursive Trees
............................280
6.5
Profile and Height of Binary Search Trees and Related Trees
.. 291
6.5.1
The Profile of Binary Search Trees and Related Trees
.. 291
6.5.2
The Height of Binary Search Trees and Related Trees
.. 300
Tries and Digital Search Trees
.............................307
7.1
The Profile of Tries
......................................308
7.1.1
Generating Functions for the Profile
.................308
7.1.2
The Expected Profile of Tries
.......................311
7.1.3
The Limiting Distribution of the Profile of Tries
.......321
7.1.4
The Height of Tries
................................323
7.1.5
Symmetric Tries
...................................324
7.2
The Profile of Digital Search Trees
.........................325
XVI Contents
7.2.1
Generating Functions for the Profile
.................325
7.2.2
The Expected Profile of Digital Search Trees
..........327
7.2.3
Symmetric Digital Search Trees
.....................337
8
Recursive Algorithms and the Contraction Method
........343
8.1
The Number of Comparisons in Quicksort
..................345
8.2
The L2 Setting of the Contraction Method
..................350
8.2.1
A General Type of Recurrence
......................350
8.2.2
A General L2 Convergence Theorem
.................352
8.2.3
Applications of the L2 Setting
......................357
8.3
Limitations of the L2 Setting and Extensions
...............361
8.3.1
The Zolotarev Metric
..............................362
8.3.2
Degenerate Limit Equations
........................363
9
Planar Graphs
.............................................365
9.1
Basic Notions
...........................................366
9.2
Counting Planar Graphs
.................................368
9.2.1
Outerplanar Graphs
...............................368
9.2.2
Series-Parallel Graphs
..............................376
9.2.3
Quadrangulations and Planar Maps
..................382
9.2.4
Planar Graphs
....................................389
9.3
Outerplanar Graphs
.....................................396
9.3.1
The Degree Distribution of Outerplanar Graphs
.......396
9.3.2
Vertices of Given Degree in Dissections
...............400
9.3.3
Vertices of Given Degree in 2-Connected Outerplanar
Graphs
...........................................404
9.3.4
Vertices of Given Degree in Connected Outerplanar
Graphs
...........................................406
9.4
Series-Parallel Graphs
....................................408
9.4.1
The Degree Distribution of Series-Parallel Graphs
.....408
9.4.2
Vertices of Given Degree in Series-Parallel Networks
... 415
9.4.3
Vertices of Given Degree in 2-Connected Series-Parallel
Graphs
...........................................416
9.4.4
Vertices of Given Degree in Connected Series-Parallel
Graphs
...........................................419
9.5
All Planar Graphs
.......................................420
9.5.1
The Degree of a Rooted Vertex
.....................421
9.5.2
Singular Expansions
...............................425
9.5.3
Degree Distribution for Planar Graphs
...............429
9.5.4
Vertices of Degree
1
or
2
in Planar Graphs
...........433
Appendix
..................................................439
References
.....................................................445
Contents XVII
Index..........................................................455
|
| adam_txt |
Contents
Classes
of Random Trees
. 1
1.1
Basic Notions
. 2
1.1.1
Rooted Versus Unrooted trees
. 2
1.1.2
Plane Versus Non-Plane trees
. 3
1.1.3
Labelled Versus Unlabelled Trees
. 3
1.2
Combinatorial Trees
. 4
1.2.1
Binary Trees
. 5
1.2.2
Planted Plane Trees
. 6
1.2.3
Labelled Trees
. 7
1.2.4
Labelled Plane Trees
. 8
1.2.5
Unlabelled Trees
. 8
1.2.6
Unlabelled Plane Trees
. 9
1.2.7
Simply Generated Trees
-
Galton-Watson Trees
. 9
1.3
Recursive Trees
. 13
1.3.1
Non-Plane Recursive Trees
. 13
1.3.2
Plane Oriented Recursive Trees
. 14
1.3.3
Increasing Trees
. 15
1.4
Search Trees
. 17
1.4.1
Binary Search Trees
. 18
1.4.2
Fringe Balanced m-Ary Search Trees
. 19
1.4.3
Digital Search Trees
. 21
1.4.4
Tries
. 22
Generating Functions
. 25
2.1
Counting with Generating Functions
. 26
2.1.1
Generating Functions and Combinatorial Constructions
27
2.1.2
Pólya's
Theory of Counting
. 33
2.1.3 Lagrange
Inversion Formula
. 36
2.2
Asymptotics with Generating Functions
. 37
2.2.1
Asymptotic Transfers
. 38
2.2.2
Functional Equations
. 43
XIV Contents
2.2.3
Asymptotic Normality and Functional Equations
. 46
2.2.4
Transfer of Singularities
. 54
2.2.5
Systems of Functional Equations
. 62
3
Advanced Tree Counting
. 69
3.1
Generating Functions and Combinatorial Trees
. 70
3.1.1
Binary and rn-ary Trees
. 70
3.1.2
Planted Plane Trees
. 71
3.1.3
Labelled Trees
. 73
3.1.4
Simply Generated Trees
-
Galton-Watson Trees
. 75
3.1.5
Unrooted Trees
. 77
3.1.6
Trees Embedded in the Plane
. 81
3.2
Additive Parameters in Trees
. 82
3.2.1
Simply Generated Trees
-
Galton-Watson Trees
. 84
3.2.2
Unrooted Trees
. 87
3.3
Patterns in Trees
. 90
3.3.1
Planted, Rooted and Unrooted Trees
. 91
3.3.2
Generating Functions for Planted Rooted Trees
. 92
3.3.3
Rooted and Unrooted Trees
. 99
3.3.4
Asymptotic Behaviour
.101
4
The Shape of Galton-Watson Trees and
Pólya
Trees
.107
4.1
The Continuum Random Tree
.108
4.1.1
Depth-First Search of a Rooted Tree
.108
4.1.2
Real Trees
.109
4.1.3
Galton-Watson Trees and the Continuum Random Tree 111
4.2
The Profile of Galton-Watson Trees
.115
4.2.1
The Distribution of the Local Time
.118
4.2.2
Weak Convergence of Continuous Stochastic Processes
. 120
4.2.3
Combinatorics on the Profile of Galton-Watson Trees
. 125
4.2.4
Asymptotic Analysis of the Main Recurrence
.126
4.2.5
Finite Dimensional Limiting Distributions
.129
4.2.6
Tightness
.134
4.2.7
The Height of Galton-Watson Trees
.139
4.2.8
Depth-First Search
.149
4.3
The Profile of
Pólya
Trees
.154
4.3.1
Combinatorial Setup
.154
4.3.2
Asymptotic Analysis of the Main Recurrence
.156
4.3.3
Finite Dimensional Limiting Distributions
.164
4.3.4
Tightness
.168
4.3.5
The Height of
Pólya
Trees
.177
Contents
XV
The Vertical Profile of Trees
.187
5.1
Quadrangulations and Embedded Trees
.188
5.2
Profiles of Trees and Random Measures
.196
5.2.1
General Profiles
.196
5.2.2
Space Embedded Trees and ISE
.196
5.2.3
The Distribution of the ISE
.204
5.3
Combinatorics on Embedded Trees
.207
5.3.1
Embedded Trees with Increments
±1 .207
5.3.2
Embedded Trees with Increments
0, ±1 .214
5.3.3
Naturally Embedded Binary Trees
.216
5.4
Asymptotics on Embedded Trees
.219
5.4.1
Trees with Small Labels
.219
5.4.2
The Number of Nodes of Given Label
.225
5.4.3
The Number of Nodes of Large Labels
.229
5.4.4
Embedded Trees with Increments
0
and
±1.235
5.4.5
Naturally Embedded Binary Trees
.235
Recursive Trees and Binary Search Trees
.237
6.1
Permutations and Trees
.238
6.1.1
Permutations and Recursive Trees
.239
6.1.2
Permutations and Binary Search Trees
.246
6.2
Generating Functions and Basic Statistics
.247
6.2.1
Generating Functions for Recursive Trees
.248
6.2.2
Generating Functions for Binary Search Trees
.249
6.2.3
Generating Functions for Plane Oriented Recursive
Trees
.251
6.2.4
The Degree Distribution of Recursive Trees
.253
6.2.5
The Insertion Depth
.262
6.3
The Profile of Recursive Trees
.265
6.3.1
The Martingale Method
.266
6.3.2
The Moment Method
.275
6.3.3
The Contraction Method
.278
6.4
The Height of Recursive Trees
.280
6.5
Profile and Height of Binary Search Trees and Related Trees
. 291
6.5.1
The Profile of Binary Search Trees and Related Trees
. 291
6.5.2
The Height of Binary Search Trees and Related Trees
. 300
Tries and Digital Search Trees
.307
7.1
The Profile of Tries
.308
7.1.1
Generating Functions for the Profile
.308
7.1.2
The Expected Profile of Tries
.311
7.1.3
The Limiting Distribution of the Profile of Tries
.321
7.1.4
The Height of Tries
.323
7.1.5
Symmetric Tries
.324
7.2
The Profile of Digital Search Trees
.325
XVI Contents
7.2.1
Generating Functions for the Profile
.325
7.2.2
The Expected Profile of Digital Search Trees
.327
7.2.3
Symmetric Digital Search Trees
.337
8
Recursive Algorithms and the Contraction Method
.343
8.1
The Number of Comparisons in Quicksort
.345
8.2
The L2 Setting of the Contraction Method
.350
8.2.1
A General Type of Recurrence
.350
8.2.2
A General L2 Convergence Theorem
.352
8.2.3
Applications of the L2 Setting
.357
8.3
Limitations of the L2 Setting and Extensions
.361
8.3.1
The Zolotarev Metric
.362
8.3.2
Degenerate Limit Equations
.363
9
Planar Graphs
.365
9.1
Basic Notions
.366
9.2
Counting Planar Graphs
.368
9.2.1
Outerplanar Graphs
.368
9.2.2
Series-Parallel Graphs
.376
9.2.3
Quadrangulations and Planar Maps
.382
9.2.4
Planar Graphs
.389
9.3
Outerplanar Graphs
.396
9.3.1
The Degree Distribution of Outerplanar Graphs
.396
9.3.2
Vertices of Given Degree in Dissections
.400
9.3.3
Vertices of Given Degree in 2-Connected Outerplanar
Graphs
.404
9.3.4
Vertices of Given Degree in Connected Outerplanar
Graphs
.406
9.4
Series-Parallel Graphs
.408
9.4.1
The Degree Distribution of Series-Parallel Graphs
.408
9.4.2
Vertices of Given Degree in Series-Parallel Networks
. 415
9.4.3
Vertices of Given Degree in 2-Connected Series-Parallel
Graphs
.416
9.4.4
Vertices of Given Degree in Connected Series-Parallel
Graphs
.419
9.5
All Planar Graphs
.420
9.5.1
The Degree of a Rooted Vertex
.421
9.5.2
Singular Expansions
.425
9.5.3
Degree Distribution for Planar Graphs
.429
9.5.4
Vertices of Degree
1
or
2
in Planar Graphs
.433
Appendix
.439
References
.445
Contents XVII
Index.455 |
| any_adam_object | 1 |
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| author | Drmota, Michael 1964- |
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| author_facet | Drmota, Michael 1964- |
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| ctrlnum | (OCoLC)305126814 (DE-599)OBVAC07007548 |
| dewey-full | 511.52 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 511 - General principles of mathematics |
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| dewey-search | 511.52 |
| dewey-sort | 3511.52 |
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| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| format | Book |
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| id | DE-604.BV035154202 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T22:48:00Z |
| indexdate | 2024-07-09T21:26:13Z |
| institution | BVB |
| isbn | 9783211753552 3211753559 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016961408 |
| oclc_num | 305126814 |
| open_access_boolean | |
| owner | DE-703 DE-384 DE-29T DE-824 DE-83 |
| owner_facet | DE-703 DE-384 DE-29T DE-824 DE-83 |
| physical | XVII, 458 Seiten Diagramme |
| publishDate | 2009 |
| publishDateSearch | 2009 |
| publishDateSort | 2009 |
| publisher | Springer |
| record_format | marc |
| spelling | Drmota, Michael 1964- (DE-588)131835718 aut Random trees an interplay between combinatorics and probability by Michael Drmota Wien ; New York Springer 2009 XVII, 458 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Out of research related to (random) trees, several asymptotic and probabilistic techniques have been developed to describe characteristics of large trees in different settings. The aim here is to provide an introduction to various aspects of trees in random settings and a systematic treatment of the involved mathematical techniques. Stochastic processes Trees (Graph theory) Graphentheorie (DE-588)4113782-6 gnd rswk-swf Zufallsgraph (DE-588)4277661-2 gnd rswk-swf Baum Mathematik (DE-588)4004849-4 gnd rswk-swf Stochastischer Prozess (DE-588)4057630-9 gnd rswk-swf Graphentheorie (DE-588)4113782-6 s Stochastischer Prozess (DE-588)4057630-9 s Zufallsgraph (DE-588)4277661-2 s Baum Mathematik (DE-588)4004849-4 s DE-604 Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016961408&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Drmota, Michael 1964- Random trees an interplay between combinatorics and probability Stochastic processes Trees (Graph theory) Graphentheorie (DE-588)4113782-6 gnd Zufallsgraph (DE-588)4277661-2 gnd Baum Mathematik (DE-588)4004849-4 gnd Stochastischer Prozess (DE-588)4057630-9 gnd |
| subject_GND | (DE-588)4113782-6 (DE-588)4277661-2 (DE-588)4004849-4 (DE-588)4057630-9 |
| title | Random trees an interplay between combinatorics and probability |
| title_auth | Random trees an interplay between combinatorics and probability |
| title_exact_search | Random trees an interplay between combinatorics and probability |
| title_exact_search_txtP | Random trees an interplay between combinatorics and probability |
| title_full | Random trees an interplay between combinatorics and probability by Michael Drmota |
| title_fullStr | Random trees an interplay between combinatorics and probability by Michael Drmota |
| title_full_unstemmed | Random trees an interplay between combinatorics and probability by Michael Drmota |
| title_short | Random trees |
| title_sort | random trees an interplay between combinatorics and probability |
| title_sub | an interplay between combinatorics and probability |
| topic | Stochastic processes Trees (Graph theory) Graphentheorie (DE-588)4113782-6 gnd Zufallsgraph (DE-588)4277661-2 gnd Baum Mathematik (DE-588)4004849-4 gnd Stochastischer Prozess (DE-588)4057630-9 gnd |
| topic_facet | Stochastic processes Trees (Graph theory) Graphentheorie Zufallsgraph Baum Mathematik Stochastischer Prozess |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016961408&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT drmotamichael randomtreesaninterplaybetweencombinatoricsandprobability |