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Asymptotic Enumeration of Compacted Binary Trees with Height - - PowerPoint PPT Presentation

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Compacted Binary Trees Asymptotic Enumeration of Compacted Binary Trees with Height Restrictions CLA 05/2018 Michael Wallner joint work with Antoine Genitrini, Bernhard Gittenberger and Manuel Kauers Erwin Schr odinger-Fellow (Austrian

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SLIDE 1

Compacted Binary Trees

Asymptotic Enumeration of Compacted Binary Trees with Height Restrictions

CLA 05/2018 Michael Wallner

joint work with Antoine Genitrini, Bernhard Gittenberger and Manuel Kauers

Erwin Schr¨

  • dinger-Fellow (Austrian Science Fund (FWF): J 4162)

Laboratoire Bordelais de Recherche en Informatique, Universit´ e de Bordeaux, France

May 24th, 2018 Based on the paper: Asymptotic Enumeration of Compacted Binary Trees, submitted to a journal. ArXiv:1703.10031

Michael Wallner | LaBRI | 24.05.2018 1 / 36

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SLIDE 2

Compacted Binary Trees | Creating a compacted tree

Creating a compacted tree

Michael Wallner | LaBRI | 24.05.2018 2 / 36

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SLIDE 3

Compacted Binary Trees | Creating a compacted tree

Motivation – Efficiently store redundant information

Example

Consider the labeled tree necessary to store the arithmetic expression (* (- (* x x) (* y y)) (+ (* x x) (* y y))) which represents (x2 − y 2)(x2 + y 2).

Michael Wallner | LaBRI | 24.05.2018 3 / 36

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SLIDE 4

Compacted Binary Trees | Creating a compacted tree

Motivation – Efficiently store redundant information

Example

Consider the labeled tree necessary to store the arithmetic expression (* (- (* x x) (* y y)) (+ (* x x) (* y y))) which represents (x2 − y 2)(x2 + y 2).

x x × y y × x x × + y y × − ×

Michael Wallner | LaBRI | 24.05.2018 3 / 36

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SLIDE 5

Compacted Binary Trees | Creating a compacted tree

Motivation – Efficiently store redundant information

Example

Consider the labeled tree necessary to store the arithmetic expression (* (- (* x x) (* y y)) (+ (* x x) (* y y))) which represents (x2 − y 2)(x2 + y 2).

x x × y y × x x × + y y × − ×

Michael Wallner | LaBRI | 24.05.2018 3 / 36

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SLIDE 6

Compacted Binary Trees | Creating a compacted tree

Motivation – Efficiently store redundant information

Example

Consider the labeled tree necessary to store the arithmetic expression (* (- (* x x) (* y y)) (+ (* x x) (* y y))) which represents (x2 − y 2)(x2 + y 2).

1 x × y y × x x × + y y × − ×

(1, (x, 0, 0))

Michael Wallner | LaBRI | 24.05.2018 3 / 36

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SLIDE 7

Compacted Binary Trees | Creating a compacted tree

Motivation – Efficiently store redundant information

Example

Consider the labeled tree necessary to store the arithmetic expression (* (- (* x x) (* y y)) (+ (* x x) (* y y))) which represents (x2 − y 2)(x2 + y 2).

1 x × y y × x x × + y y × − ×

(1, (x, 0, 0))

Michael Wallner | LaBRI | 24.05.2018 3 / 36

slide-8
SLIDE 8

Compacted Binary Trees | Creating a compacted tree

Motivation – Efficiently store redundant information

Example

Consider the labeled tree necessary to store the arithmetic expression (* (- (* x x) (* y y)) (+ (* x x) (* y y))) which represents (x2 − y 2)(x2 + y 2).

1 1 × y y × x x × + y y × − ×

(1, (x, 0, 0))

Michael Wallner | LaBRI | 24.05.2018 3 / 36

slide-9
SLIDE 9

Compacted Binary Trees | Creating a compacted tree

Motivation – Efficiently store redundant information

Example

Consider the labeled tree necessary to store the arithmetic expression (* (- (* x x) (* y y)) (+ (* x x) (* y y))) which represents (x2 − y 2)(x2 + y 2).

1 1 × y y × x x × + y y × − ×

(1, (x, 0, 0))

Michael Wallner | LaBRI | 24.05.2018 3 / 36

slide-10
SLIDE 10

Compacted Binary Trees | Creating a compacted tree

Motivation – Efficiently store redundant information

Example

Consider the labeled tree necessary to store the arithmetic expression (* (- (* x x) (* y y)) (+ (* x x) (* y y))) which represents (x2 − y 2)(x2 + y 2).

1 1 2 y y × x x × + y y × − ×

(1, (x, 0, 0)), (2, (×, 1, 1))

Michael Wallner | LaBRI | 24.05.2018 3 / 36

slide-11
SLIDE 11

Compacted Binary Trees | Creating a compacted tree

Motivation – Efficiently store redundant information

Example

Consider the labeled tree necessary to store the arithmetic expression (* (- (* x x) (* y y)) (+ (* x x) (* y y))) which represents (x2 − y 2)(x2 + y 2).

1 1 2 y y × x x × + y y × − ×

(1, (x, 0, 0)), (2, (×, 1, 1))

Michael Wallner | LaBRI | 24.05.2018 3 / 36

slide-12
SLIDE 12

Compacted Binary Trees | Creating a compacted tree

Motivation – Efficiently store redundant information

Example

Consider the labeled tree necessary to store the arithmetic expression (* (- (* x x) (* y y)) (+ (* x x) (* y y))) which represents (x2 − y 2)(x2 + y 2).

1 1 2 3 y × x x × + y y × − ×

(1, (x, 0, 0)), (2, (×, 1, 1)), (3, (y, 0, 0))

Michael Wallner | LaBRI | 24.05.2018 3 / 36

slide-13
SLIDE 13

Compacted Binary Trees | Creating a compacted tree

Motivation – Efficiently store redundant information

Example

Consider the labeled tree necessary to store the arithmetic expression (* (- (* x x) (* y y)) (+ (* x x) (* y y))) which represents (x2 − y 2)(x2 + y 2).

1 1 2 3 y × x x × + y y × − ×

(1, (x, 0, 0)), (2, (×, 1, 1)), (3, (y, 0, 0))

Michael Wallner | LaBRI | 24.05.2018 3 / 36

slide-14
SLIDE 14

Compacted Binary Trees | Creating a compacted tree

Motivation – Efficiently store redundant information

Example

Consider the labeled tree necessary to store the arithmetic expression (* (- (* x x) (* y y)) (+ (* x x) (* y y))) which represents (x2 − y 2)(x2 + y 2).

1 1 2 3 3 × x x × + y y × − ×

(1, (x, 0, 0)), (2, (×, 1, 1)), (3, (y, 0, 0))

Michael Wallner | LaBRI | 24.05.2018 3 / 36

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SLIDE 15

Compacted Binary Trees | Creating a compacted tree

Motivation – Efficiently store redundant information

Example

Consider the labeled tree necessary to store the arithmetic expression (* (- (* x x) (* y y)) (+ (* x x) (* y y))) which represents (x2 − y 2)(x2 + y 2).

1 1 2 3 3 × x x × + y y × − ×

(1, (x, 0, 0)), (2, (×, 1, 1)), (3, (y, 0, 0))

Michael Wallner | LaBRI | 24.05.2018 3 / 36

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SLIDE 16

Compacted Binary Trees | Creating a compacted tree

Motivation – Efficiently store redundant information

Example

Consider the labeled tree necessary to store the arithmetic expression (* (- (* x x) (* y y)) (+ (* x x) (* y y))) which represents (x2 − y 2)(x2 + y 2).

1 1 2 3 3 x x × + y y × − × 4

(1, (x, 0, 0)), (2, (×, 1, 1)), (3, (y, 0, 0)), (4, (×, 3, 3))

Michael Wallner | LaBRI | 24.05.2018 3 / 36

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SLIDE 17

Compacted Binary Trees | Creating a compacted tree

Motivation – Efficiently store redundant information

Example

Consider the labeled tree necessary to store the arithmetic expression (* (- (* x x) (* y y)) (+ (* x x) (* y y))) which represents (x2 − y 2)(x2 + y 2).

1 1 2 3 3 x x × + y y × − × 4

(1, (x, 0, 0)), (2, (×, 1, 1)), (3, (y, 0, 0)), (4, (×, 3, 3))

Michael Wallner | LaBRI | 24.05.2018 3 / 36

slide-18
SLIDE 18

Compacted Binary Trees | Creating a compacted tree

Motivation – Efficiently store redundant information

Example

Consider the labeled tree necessary to store the arithmetic expression (* (- (* x x) (* y y)) (+ (* x x) (* y y))) which represents (x2 − y 2)(x2 + y 2).

1 1 2 3 3 x x × + y y × × 4 5

(1, (x, 0, 0)), (2, (×, 1, 1)), (3, (y, 0, 0)), (4, (×, 3, 3)), (5, (−, 2, 4))

Michael Wallner | LaBRI | 24.05.2018 3 / 36

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SLIDE 19

Compacted Binary Trees | Creating a compacted tree

Motivation – Efficiently store redundant information

Example

Consider the labeled tree necessary to store the arithmetic expression (* (- (* x x) (* y y)) (+ (* x x) (* y y))) which represents (x2 − y 2)(x2 + y 2).

1 1 2 3 3 x x × + y y × × 4 5

(1, (x, 0, 0)), (2, (×, 1, 1)), (3, (y, 0, 0)), (4, (×, 3, 3)), (5, (−, 2, 4))

Michael Wallner | LaBRI | 24.05.2018 3 / 36

slide-20
SLIDE 20

Compacted Binary Trees | Creating a compacted tree

Motivation – Efficiently store redundant information

Example

Consider the labeled tree necessary to store the arithmetic expression (* (- (* x x) (* y y)) (+ (* x x) (* y y))) which represents (x2 − y 2)(x2 + y 2).

1 1 2 3 3 1 x × + y y × × 4 5

(1, (x, 0, 0)), (2, (×, 1, 1)), (3, (y, 0, 0)), (4, (×, 3, 3)), (5, (−, 2, 4))

Michael Wallner | LaBRI | 24.05.2018 3 / 36

slide-21
SLIDE 21

Compacted Binary Trees | Creating a compacted tree

Motivation – Efficiently store redundant information

Example

Consider the labeled tree necessary to store the arithmetic expression (* (- (* x x) (* y y)) (+ (* x x) (* y y))) which represents (x2 − y 2)(x2 + y 2).

1 1 2 3 3 1 x × + y y × × 4 5

(1, (x, 0, 0)), (2, (×, 1, 1)), (3, (y, 0, 0)), (4, (×, 3, 3)), (5, (−, 2, 4))

Michael Wallner | LaBRI | 24.05.2018 3 / 36

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SLIDE 22

Compacted Binary Trees | Creating a compacted tree

Motivation – Efficiently store redundant information

Example

Consider the labeled tree necessary to store the arithmetic expression (* (- (* x x) (* y y)) (+ (* x x) (* y y))) which represents (x2 − y 2)(x2 + y 2).

1 1 2 3 3 1 1 × + y y × × 4 5

(1, (x, 0, 0)), (2, (×, 1, 1)), (3, (y, 0, 0)), (4, (×, 3, 3)), (5, (−, 2, 4))

Michael Wallner | LaBRI | 24.05.2018 3 / 36

slide-23
SLIDE 23

Compacted Binary Trees | Creating a compacted tree

Motivation – Efficiently store redundant information

Example

Consider the labeled tree necessary to store the arithmetic expression (* (- (* x x) (* y y)) (+ (* x x) (* y y))) which represents (x2 − y 2)(x2 + y 2).

1 1 2 3 3 1 1 × + y y × × 4 5

(1, (x, 0, 0)), (2, (×, 1, 1)), (3, (y, 0, 0)), (4, (×, 3, 3)), (5, (−, 2, 4))

Michael Wallner | LaBRI | 24.05.2018 3 / 36

slide-24
SLIDE 24

Compacted Binary Trees | Creating a compacted tree

Motivation – Efficiently store redundant information

Example

Consider the labeled tree necessary to store the arithmetic expression (* (- (* x x) (* y y)) (+ (* x x) (* y y))) which represents (x2 − y 2)(x2 + y 2).

1 1 2 3 3 1 1 + y y × × 4 5 2

(1, (x, 0, 0)), (2, (×, 1, 1)), (3, (y, 0, 0)), (4, (×, 3, 3)), (5, (−, 2, 4))

Michael Wallner | LaBRI | 24.05.2018 3 / 36

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SLIDE 25

Compacted Binary Trees | Creating a compacted tree

Motivation – Efficiently store redundant information

Example

Consider the labeled tree necessary to store the arithmetic expression (* (- (* x x) (* y y)) (+ (* x x) (* y y))) which represents (x2 − y 2)(x2 + y 2).

1 1 2 3 3 1 1 + y y × × 4 5 2

(1, (x, 0, 0)), (2, (×, 1, 1)), (3, (y, 0, 0)), (4, (×, 3, 3)), (5, (−, 2, 4))

Michael Wallner | LaBRI | 24.05.2018 3 / 36

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SLIDE 26

Compacted Binary Trees | Creating a compacted tree

Motivation – Efficiently store redundant information

Example

Consider the labeled tree necessary to store the arithmetic expression (* (- (* x x) (* y y)) (+ (* x x) (* y y))) which represents (x2 − y 2)(x2 + y 2).

1 1 2 3 3 1 1 + 3 y × × 4 5 2

(1, (x, 0, 0)), (2, (×, 1, 1)), (3, (y, 0, 0)), (4, (×, 3, 3)), (5, (−, 2, 4))

Michael Wallner | LaBRI | 24.05.2018 3 / 36

slide-27
SLIDE 27

Compacted Binary Trees | Creating a compacted tree

Motivation – Efficiently store redundant information

Example

Consider the labeled tree necessary to store the arithmetic expression (* (- (* x x) (* y y)) (+ (* x x) (* y y))) which represents (x2 − y 2)(x2 + y 2).

1 1 2 3 3 1 1 + 3 y × × 4 5 2

(1, (x, 0, 0)), (2, (×, 1, 1)), (3, (y, 0, 0)), (4, (×, 3, 3)), (5, (−, 2, 4))

Michael Wallner | LaBRI | 24.05.2018 3 / 36

slide-28
SLIDE 28

Compacted Binary Trees | Creating a compacted tree

Motivation – Efficiently store redundant information

Example

Consider the labeled tree necessary to store the arithmetic expression (* (- (* x x) (* y y)) (+ (* x x) (* y y))) which represents (x2 − y 2)(x2 + y 2).

1 1 2 3 3 1 1 + 3 3 × × 4 5 2

(1, (x, 0, 0)), (2, (×, 1, 1)), (3, (y, 0, 0)), (4, (×, 3, 3)), (5, (−, 2, 4))

Michael Wallner | LaBRI | 24.05.2018 3 / 36

slide-29
SLIDE 29

Compacted Binary Trees | Creating a compacted tree

Motivation – Efficiently store redundant information

Example

Consider the labeled tree necessary to store the arithmetic expression (* (- (* x x) (* y y)) (+ (* x x) (* y y))) which represents (x2 − y 2)(x2 + y 2).

1 1 2 3 3 1 1 + 3 3 × × 4 5 2

(1, (x, 0, 0)), (2, (×, 1, 1)), (3, (y, 0, 0)), (4, (×, 3, 3)), (5, (−, 2, 4))

Michael Wallner | LaBRI | 24.05.2018 3 / 36

slide-30
SLIDE 30

Compacted Binary Trees | Creating a compacted tree

Motivation – Efficiently store redundant information

Example

Consider the labeled tree necessary to store the arithmetic expression (* (- (* x x) (* y y)) (+ (* x x) (* y y))) which represents (x2 − y 2)(x2 + y 2).

1 1 2 3 3 1 1 + 3 3 × 4 5 2 4

(1, (x, 0, 0)), (2, (×, 1, 1)), (3, (y, 0, 0)), (4, (×, 3, 3)), (5, (−, 2, 4))

Michael Wallner | LaBRI | 24.05.2018 3 / 36

slide-31
SLIDE 31

Compacted Binary Trees | Creating a compacted tree

Motivation – Efficiently store redundant information

Example

Consider the labeled tree necessary to store the arithmetic expression (* (- (* x x) (* y y)) (+ (* x x) (* y y))) which represents (x2 − y 2)(x2 + y 2).

1 1 2 3 3 1 1 + 3 3 × 4 5 2 4

(1, (x, 0, 0)), (2, (×, 1, 1)), (3, (y, 0, 0)), (4, (×, 3, 3)), (5, (−, 2, 4))

Michael Wallner | LaBRI | 24.05.2018 3 / 36

slide-32
SLIDE 32

Compacted Binary Trees | Creating a compacted tree

Motivation – Efficiently store redundant information

Example

Consider the labeled tree necessary to store the arithmetic expression (* (- (* x x) (* y y)) (+ (* x x) (* y y))) which represents (x2 − y 2)(x2 + y 2).

1 1 2 3 3 1 1 3 3 × 4 5 2 4 6

(1, (x, 0, 0)), (2, (×, 1, 1)), (3, (y, 0, 0)), (4, (×, 3, 3)), (5, (−, 2, 4)), (6, (−, 2, 4))

Michael Wallner | LaBRI | 24.05.2018 3 / 36

slide-33
SLIDE 33

Compacted Binary Trees | Creating a compacted tree

Motivation – Efficiently store redundant information

Example

Consider the labeled tree necessary to store the arithmetic expression (* (- (* x x) (* y y)) (+ (* x x) (* y y))) which represents (x2 − y 2)(x2 + y 2).

1 1 2 3 3 1 1 3 3 × 4 5 2 4 6

(1, (x, 0, 0)), (2, (×, 1, 1)), (3, (y, 0, 0)), (4, (×, 3, 3)), (5, (−, 2, 4)), (6, (−, 2, 4))

Michael Wallner | LaBRI | 24.05.2018 3 / 36

slide-34
SLIDE 34

Compacted Binary Trees | Creating a compacted tree

Motivation – Efficiently store redundant information

Example

Consider the labeled tree necessary to store the arithmetic expression (* (- (* x x) (* y y)) (+ (* x x) (* y y))) which represents (x2 − y 2)(x2 + y 2).

1 1 2 3 3 1 1 3 3 4 5 2 4 6 7

(1, (x, 0, 0)), (2, (×, 1, 1)), (3, (y, 0, 0)), (4, (×, 3, 3)), (5, (−, 2, 4)), (6, (−, 2, 4)), (7, (−, 5, 6))

Michael Wallner | LaBRI | 24.05.2018 3 / 36

slide-35
SLIDE 35

Compacted Binary Trees | Creating a compacted tree

Motivation – Efficiently store redundant information

Example

Consider the labeled tree necessary to store the arithmetic expression (* (- (* x x) (* y y)) (+ (* x x) (* y y))) which represents (x2 − y 2)(x2 + y 2).

1 1 2 3 3 1 1 3 3 4 5 2 4 6 7 x × y × + − ×

(1, (x, 0, 0)), (2, (×, 1, 1)), (3, (y, 0, 0)), (4, (×, 3, 3)), (5, (−, 2, 4)), (6, (−, 2, 4)), (7, (−, 5, 6))

Michael Wallner | LaBRI | 24.05.2018 3 / 36

slide-36
SLIDE 36

Compacted Binary Trees | Creating a compacted tree

Motivation – Efficiently store redundant information

Example

Consider the labeled tree necessary to store the arithmetic expression (* (- (* x x) (* y y)) (+ (* x x) (* y y))) which represents (x2 − y 2)(x2 + y 2).

1 1 2 3 3 1 1 3 3 4 5 2 4 6 7 x × y × + − ×

(1, (x, 0, 0)), (2, (×, 1, 1)), (3, (y, 0, 0)), (4, (×, 3, 3)), (5, (−, 2, 4)), (6, (−, 2, 4)), (7, (−, 5, 6))

Definition

Compacted tree is the DAG computed by this procedure.

Michael Wallner | LaBRI | 24.05.2018 3 / 36

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SLIDE 37

Compacted Binary Trees | Creating a compacted tree

Compacted trees

Important property: Subtrees are unique Efficient algorithm to compute compacted tree

Traverse tree post-order If subtree appears twice, delete second one and replace by pointer → directed acyclic graph (DAG)

Analyzed by [Flajolet, Sipala, Steyaert 1990]: A tree of size n has a compacted form of expected size that is asymptotically equal to C n √log n, where C is explicit related to the type of trees and the statistical model. Applications: XML-Compression [Bousquet-M´ elou, Lohrey, Maneth, Noeth 2015], Compilers [Aho, Sethi, Ullman 1986], LISP [Goto 1974], Data storage [Meinel, Theobald 1998], [Knuth 1968], etc. Restrict to unlabeled binary trees

Michael Wallner | LaBRI | 24.05.2018 4 / 36

slide-38
SLIDE 38

Compacted Binary Trees | Creating a compacted tree

Compacted trees

Important property: Subtrees are unique Efficient algorithm to compute compacted tree

Traverse tree post-order If subtree appears twice, delete second one and replace by pointer → directed acyclic graph (DAG)

Analyzed by [Flajolet, Sipala, Steyaert 1990]: A tree of size n has a compacted form of expected size that is asymptotically equal to C n √log n, where C is explicit related to the type of trees and the statistical model. Applications: XML-Compression [Bousquet-M´ elou, Lohrey, Maneth, Noeth 2015], Compilers [Aho, Sethi, Ullman 1986], LISP [Goto 1974], Data storage [Meinel, Theobald 1998], [Knuth 1968], etc. Restrict to unlabeled binary trees

Michael Wallner | LaBRI | 24.05.2018 4 / 36

slide-39
SLIDE 39

Compacted Binary Trees | Creating a compacted tree

Compacted trees

Important property: Subtrees are unique Efficient algorithm to compute compacted tree

Traverse tree post-order If subtree appears twice, delete second one and replace by pointer → directed acyclic graph (DAG)

Analyzed by [Flajolet, Sipala, Steyaert 1990]: A tree of size n has a compacted form of expected size that is asymptotically equal to C n √log n, where C is explicit related to the type of trees and the statistical model. Applications: XML-Compression [Bousquet-M´ elou, Lohrey, Maneth, Noeth 2015], Compilers [Aho, Sethi, Ullman 1986], LISP [Goto 1974], Data storage [Meinel, Theobald 1998], [Knuth 1968], etc. Restrict to unlabeled binary trees

Michael Wallner | LaBRI | 24.05.2018 4 / 36

slide-40
SLIDE 40

Compacted Binary Trees | Creating a compacted tree

Compacted trees

Important property: Subtrees are unique Efficient algorithm to compute compacted tree

Traverse tree post-order If subtree appears twice, delete second one and replace by pointer → directed acyclic graph (DAG)

Analyzed by [Flajolet, Sipala, Steyaert 1990]: A tree of size n has a compacted form of expected size that is asymptotically equal to C n √log n, where C is explicit related to the type of trees and the statistical model. Applications: XML-Compression [Bousquet-M´ elou, Lohrey, Maneth, Noeth 2015], Compilers [Aho, Sethi, Ullman 1986], LISP [Goto 1974], Data storage [Meinel, Theobald 1998], [Knuth 1968], etc. Restrict to unlabeled binary trees

Michael Wallner | LaBRI | 24.05.2018 4 / 36

slide-41
SLIDE 41

Compacted Binary Trees | Creating a compacted tree

Compacted trees

Important property: Subtrees are unique Efficient algorithm to compute compacted tree

Traverse tree post-order If subtree appears twice, delete second one and replace by pointer → directed acyclic graph (DAG)

Analyzed by [Flajolet, Sipala, Steyaert 1990]: A tree of size n has a compacted form of expected size that is asymptotically equal to C n √log n, where C is explicit related to the type of trees and the statistical model. Applications: XML-Compression [Bousquet-M´ elou, Lohrey, Maneth, Noeth 2015], Compilers [Aho, Sethi, Ullman 1986], LISP [Goto 1974], Data storage [Meinel, Theobald 1998], [Knuth 1968], etc. Restrict to unlabeled binary trees

Michael Wallner | LaBRI | 24.05.2018 4 / 36

slide-42
SLIDE 42

Compacted Binary Trees | Creating a compacted tree

Compacted trees

Important property: Subtrees are unique Efficient algorithm to compute compacted tree

Traverse tree post-order If subtree appears twice, delete second one and replace by pointer → directed acyclic graph (DAG)

Analyzed by [Flajolet, Sipala, Steyaert 1990]: A tree of size n has a compacted form of expected size that is asymptotically equal to C n √log n, where C is explicit related to the type of trees and the statistical model. Applications: XML-Compression [Bousquet-M´ elou, Lohrey, Maneth, Noeth 2015], Compilers [Aho, Sethi, Ullman 1986], LISP [Goto 1974], Data storage [Meinel, Theobald 1998], [Knuth 1968], etc. Restrict to unlabeled binary trees

Reverse question

How many compacted trees of (compacted) size n exist?

Michael Wallner | LaBRI | 24.05.2018 4 / 36

slide-43
SLIDE 43

Compacted Binary Trees | Creating a compacted tree

Compacted trees

Size of a compacted tree: number of internal nodes Number of compacted trees of size n: cn

Michael Wallner | LaBRI | 24.05.2018 5 / 36

slide-44
SLIDE 44

Compacted Binary Trees | Creating a compacted tree

Compacted trees

Size of a compacted tree: number of internal nodes Number of compacted trees of size n: cn

Michael Wallner | LaBRI | 24.05.2018 5 / 36

slide-45
SLIDE 45

Compacted Binary Trees | Creating a compacted tree

Compacted trees

Size of a compacted tree: number of internal nodes Number of compacted trees of size n: cn

Figure: All compacted binary trees of size n = 0, 1, 2.

Michael Wallner | LaBRI | 24.05.2018 5 / 36

slide-46
SLIDE 46

Compacted Binary Trees | Creating a compacted tree

Compacted trees

Size of a compacted tree: number of internal nodes Number of compacted trees of size n: cn

Figure: All compacted binary trees of size n = 0, 1, 2.

Example (Compacted binary trees)

size n = 0 n = 1 n = 2 n = 3 n = 4 n = 5 n = 6 cn 1 1 3 15 111 1119 14487 n! ≤ cn ≤ 1 n + 1 2n n

  • · n!

Hence, cn = O(n!4nn−1/2).

Michael Wallner | LaBRI | 24.05.2018 5 / 36

slide-47
SLIDE 47

Compacted Binary Trees | Creating a compacted tree

Goals of this talk

Goals

1 Understand compacted trees 2 Find a recurrence relation for compacted trees 3 Use exponential generating functions to count DAGs 4 Solve the (simplified) problem(s)

Michael Wallner | LaBRI | 24.05.2018 6 / 36

slide-48
SLIDE 48

Compacted Binary Trees | Creating a compacted tree

Goals of this talk

Goals

1 Understand compacted trees 2 Find a recurrence relation for compacted trees 3 Use exponential generating functions to count DAGs 4 Solve the (simplified) problem(s)

Michael Wallner | LaBRI | 24.05.2018 6 / 36

slide-49
SLIDE 49

Compacted Binary Trees | Creating a compacted tree

Goals of this talk

Goals

1 Understand compacted trees 2 Find a recurrence relation for compacted trees 3 Use exponential generating functions to count DAGs 4 Solve the (simplified) problem(s)

Michael Wallner | LaBRI | 24.05.2018 6 / 36

slide-50
SLIDE 50

Compacted Binary Trees | Creating a compacted tree

Goals of this talk

Goals

1 Understand compacted trees 2 Find a recurrence relation for compacted trees 3 Use exponential generating functions to count DAGs 4 Solve the (simplified) problem(s)

Michael Wallner | LaBRI | 24.05.2018 6 / 36

slide-51
SLIDE 51

Compacted Binary Trees | Creating a compacted tree

Goals of this talk

Goals

1 Understand compacted trees 2 Find a recurrence relation for compacted trees 3 Use exponential generating functions to count DAGs 4 Solve the (simplified) problem(s)

Methods

1 Recurrence relations 2 Bijections 3 Generating functions 4 Symbolic method 5 Differential equations 6 Singularity analysis 7 Chebyshev polynomials 8 Guess and prove

Michael Wallner | LaBRI | 24.05.2018 6 / 36

slide-52
SLIDE 52

Compacted Binary Trees | Creating a compacted tree

Building a compacted tree from a binary tree

Idea

Every compacted tree of size n can be build from a binary tree of size n by adding pointers.

Michael Wallner | LaBRI | 24.05.2018 7 / 36

slide-53
SLIDE 53

Compacted Binary Trees | Creating a compacted tree

Building a compacted tree from a binary tree

Idea

Every compacted tree of size n can be build from a binary tree of size n by adding pointers. Attention: Pointers are not allowed to violate uniqueness Observation: Only cherries (nodes with 2 pointers) might violate uniqueness

Michael Wallner | LaBRI | 24.05.2018 7 / 36

slide-54
SLIDE 54

Compacted Binary Trees | Creating a compacted tree

Building a compacted tree from a binary tree

Idea

Every compacted tree of size n can be build from a binary tree of size n by adding pointers. Attention: Pointers are not allowed to violate uniqueness Observation: Only cherries (nodes with 2 pointers) might violate uniqueness

Michael Wallner | LaBRI | 24.05.2018 7 / 36

slide-55
SLIDE 55

Compacted Binary Trees | Creating a compacted tree

Building a compacted tree from a binary tree

Idea

Every compacted tree of size n can be build from a binary tree of size n by adding pointers. Attention: Pointers are not allowed to violate uniqueness Observation: Only cherries (nodes with 2 pointers) might violate uniqueness

Procedure

1 Take a binary tree of size n 2 Add leaf as left child on first free spot in post-order traversal 3 Add pointers such that out-degree of all internal nodes is 2 4 Connect pointers to leaf or to internal nodes before the root in post-order

NOT violating uniqueness

Michael Wallner | LaBRI | 24.05.2018 7 / 36

slide-56
SLIDE 56

Compacted Binary Trees | Creating a compacted tree

Building a compacted tree from a binary tree

Idea

Every compacted tree of size n can be build from a binary tree of size n by adding pointers. Attention: Pointers are not allowed to violate uniqueness Observation: Only cherries (nodes with 2 pointers) might violate uniqueness

Procedure

1 Take a binary tree of size n 2 Add leaf as left child on first free spot in post-order traversal 3 Add pointers such that out-degree of all internal nodes is 2 4 Connect pointers to leaf or to internal nodes before the root in post-order

NOT violating uniqueness

Michael Wallner | LaBRI | 24.05.2018 7 / 36

slide-57
SLIDE 57

Compacted Binary Trees | Creating a compacted tree

Building a compacted tree from a binary tree

Idea

Every compacted tree of size n can be build from a binary tree of size n by adding pointers. Attention: Pointers are not allowed to violate uniqueness Observation: Only cherries (nodes with 2 pointers) might violate uniqueness

Procedure

1 Take a binary tree of size n 2 Add leaf as left child on first free spot in post-order traversal 3 Add pointers such that out-degree of all internal nodes is 2 4 Connect pointers to leaf or to internal nodes before the root in post-order

NOT violating uniqueness

Michael Wallner | LaBRI | 24.05.2018 7 / 36

slide-58
SLIDE 58

Compacted Binary Trees | Creating a compacted tree

Building a compacted tree from a binary tree

Idea

Every compacted tree of size n can be build from a binary tree of size n by adding pointers. Attention: Pointers are not allowed to violate uniqueness Observation: Only cherries (nodes with 2 pointers) might violate uniqueness

Procedure

1 Take a binary tree of size n 2 Add leaf as left child on first free spot in post-order traversal 3 Add pointers such that out-degree of all internal nodes is 2 4 Connect pointers to leaf or to internal nodes before the root in post-order

NOT violating uniqueness

Michael Wallner | LaBRI | 24.05.2018 7 / 36

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SLIDE 59

Compacted Binary Trees | Creating a compacted tree

Building a compacted tree from a binary tree – Example

Procedure

1 Take a binary tree of size n (called spine) 2 Add leaf as left child on first free spot in post-order traversal 3 Add pointers such that out-degree of all internal nodes is 2 4 Connect pointers to leaf or to internal nodes before the root in post-order

NOT violating uniqueness

Michael Wallner | LaBRI | 24.05.2018 8 / 36

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SLIDE 60

Compacted Binary Trees | Creating a compacted tree

Building a compacted tree from a binary tree – Example

Procedure

1 Take a binary tree of size n (called spine) 2 Add leaf as left child on first free spot in post-order traversal 3 Add pointers such that out-degree of all internal nodes is 2 4 Connect pointers to leaf or to internal nodes before the root in post-order

NOT violating uniqueness

Michael Wallner | LaBRI | 24.05.2018 8 / 36

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SLIDE 61

Compacted Binary Trees | Creating a compacted tree

Building a compacted tree from a binary tree – Example

Procedure

1 Take a binary tree of size n (called spine) 2 Add leaf as left child on first free spot in post-order traversal 3 Add pointers such that out-degree of all internal nodes is 2 4 Connect pointers to leaf or to internal nodes before the root in post-order

NOT violating uniqueness

Michael Wallner | LaBRI | 24.05.2018 8 / 36

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SLIDE 62

Compacted Binary Trees | Creating a compacted tree

Building a compacted tree from a binary tree – Example

Procedure

1 Take a binary tree of size n (called spine) 2 Add leaf as left child on first free spot in post-order traversal 3 Add pointers such that out-degree of all internal nodes is 2 4 Connect pointers to leaf or to internal nodes before the root in post-order

NOT violating uniqueness

Michael Wallner | LaBRI | 24.05.2018 8 / 36

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SLIDE 63

Compacted Binary Trees | Creating a compacted tree

Building a compacted tree from a binary tree – Example

Procedure

1 Take a binary tree of size n (called spine) 2 Add leaf as left child on first free spot in post-order traversal 3 Add pointers such that out-degree of all internal nodes is 2 4 Connect pointers to leaf or to internal nodes before the root in post-order

NOT violating uniqueness

Michael Wallner | LaBRI | 24.05.2018 8 / 36

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SLIDE 64

Compacted Binary Trees | Creating a compacted tree

Building a compacted tree from a binary tree – Example

Procedure

1 Take a binary tree of size n (called spine) 2 Add leaf as left child on first free spot in post-order traversal 3 Add pointers such that out-degree of all internal nodes is 2 4 Connect pointers to leaf or to internal nodes before the root in post-order

NOT violating uniqueness Valid compacted tree

Michael Wallner | LaBRI | 24.05.2018 8 / 36

slide-65
SLIDE 65

Compacted Binary Trees | Creating a compacted tree

Building a compacted tree from a binary tree – Example

Procedure

1 Take a binary tree of size n (called spine) 2 Add leaf as left child on first free spot in post-order traversal 3 Add pointers such that out-degree of all internal nodes is 2 4 Connect pointers to leaf or to internal nodes before the root in post-order

NOT violating uniqueness Valid compacted tree

Michael Wallner | LaBRI | 24.05.2018 8 / 36

slide-66
SLIDE 66

Compacted Binary Trees | Creating a compacted tree

Building a compacted tree from a binary tree – Example

Procedure

1 Take a binary tree of size n (called spine) 2 Add leaf as left child on first free spot in post-order traversal 3 Add pointers such that out-degree of all internal nodes is 2 4 Connect pointers to leaf or to internal nodes before the root in post-order

NOT violating uniqueness Valid compacted tree

Michael Wallner | LaBRI | 24.05.2018 8 / 36

slide-67
SLIDE 67

Compacted Binary Trees | Creating a compacted tree

Building a compacted tree from a binary tree – Example

Procedure

1 Take a binary tree of size n (called spine) 2 Add leaf as left child on first free spot in post-order traversal 3 Add pointers such that out-degree of all internal nodes is 2 4 Connect pointers to leaf or to internal nodes before the root in post-order

NOT violating uniqueness Valid compacted tree

Michael Wallner | LaBRI | 24.05.2018 8 / 36

slide-68
SLIDE 68

Compacted Binary Trees | Creating a compacted tree

Building a compacted tree from a binary tree – Example

Procedure

1 Take a binary tree of size n (called spine) 2 Add leaf as left child on first free spot in post-order traversal 3 Add pointers such that out-degree of all internal nodes is 2 4 Connect pointers to leaf or to internal nodes before the root in post-order

NOT violating uniqueness Valid compacted tree

Michael Wallner | LaBRI | 24.05.2018 8 / 36

slide-69
SLIDE 69

Compacted Binary Trees | Creating a compacted tree

Building a compacted tree from a binary tree – Example

Procedure

1 Take a binary tree of size n (called spine) 2 Add leaf as left child on first free spot in post-order traversal 3 Add pointers such that out-degree of all internal nodes is 2 4 Connect pointers to leaf or to internal nodes before the root in post-order

NOT violating uniqueness Valid compacted tree Invalid compacted tree

Michael Wallner | LaBRI | 24.05.2018 8 / 36

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SLIDE 70

Compacted Binary Trees | Creating a compacted tree

Building a compacted tree from a binary tree – Example

Procedure

1 Take a binary tree of size n (called spine) 2 Add leaf as left child on first free spot in post-order traversal 3 Add pointers such that out-degree of all internal nodes is 2 4 Connect pointers to leaf or to internal nodes before the root in post-order

NOT violating uniqueness Valid compacted tree Invalid compacted tree We call the underlying binary tree from step 1 the spine.

Michael Wallner | LaBRI | 24.05.2018 8 / 36

slide-71
SLIDE 71

Compacted Binary Trees | Creating a compacted tree

Building a compacted tree from a binary tree – Example

Procedure

1 Take a binary tree of size n (called spine) 2 Add leaf as left child on first free spot in post-order traversal 3 Add pointers such that out-degree of all internal nodes is 2 4 Connect pointers to leaf or to internal nodes before the root in post-order

NOT violating uniqueness Valid compacted tree Invalid compacted tree We call the underlying binary tree from step 1 the spine. This spine is associated to 3 valid compacted trees.

Michael Wallner | LaBRI | 24.05.2018 8 / 36

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SLIDE 72

Compacted Binary Trees | Creating a compacted tree

A bigger example

We take a binary tree of size 8.

Michael Wallner | LaBRI | 24.05.2018 9 / 36

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SLIDE 73

Compacted Binary Trees | Creating a compacted tree

A bigger example

We take a binary tree of size 8.

Michael Wallner | LaBRI | 24.05.2018 9 / 36

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SLIDE 74

Compacted Binary Trees | Creating a compacted tree

A bigger example

We take a binary tree of size 8.

1

Michael Wallner | LaBRI | 24.05.2018 9 / 36

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SLIDE 75

Compacted Binary Trees | Creating a compacted tree

A bigger example

We take a binary tree of size 8.

1 2 2

Michael Wallner | LaBRI | 24.05.2018 9 / 36

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SLIDE 76

Compacted Binary Trees | Creating a compacted tree

A bigger example

We take a binary tree of size 8.

1 2 2 3

Michael Wallner | LaBRI | 24.05.2018 9 / 36

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SLIDE 77

Compacted Binary Trees | Creating a compacted tree

A bigger example

We take a binary tree of size 8.

1 2 2 3 4

Michael Wallner | LaBRI | 24.05.2018 9 / 36

slide-78
SLIDE 78

Compacted Binary Trees | Creating a compacted tree

A bigger example

We take a binary tree of size 8.

1 2 2 3 4 4 4

Michael Wallner | LaBRI | 24.05.2018 9 / 36

slide-79
SLIDE 79

Compacted Binary Trees | Creating a compacted tree

A bigger example

We take a binary tree of size 8.

1 2 2 3 4 4 4 13

Michael Wallner | LaBRI | 24.05.2018 9 / 36

slide-80
SLIDE 80

Compacted Binary Trees | Creating a compacted tree

A bigger example

We take a binary tree of size 8.

1 2 2 3 4 4 4 13 6 6

Michael Wallner | LaBRI | 24.05.2018 9 / 36

slide-81
SLIDE 81

Compacted Binary Trees | Creating a compacted tree

A bigger example

We take a binary tree of size 8.

1 2 2 3 4 4 4 13 6 6 31

Michael Wallner | LaBRI | 24.05.2018 9 / 36

slide-82
SLIDE 82

Compacted Binary Trees | Creating a compacted tree

A bigger example

We take a binary tree of size 8.

1 2 2 3 4 4 4 13 6 6 31

In total, this spine corresponds to 1 · 3 · 4 · 13 · 31 = 4836 compacted trees.

Michael Wallner | LaBRI | 24.05.2018 9 / 36

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SLIDE 83

Compacted Binary Trees | A recurrence relation

A recurrence relation

Michael Wallner | LaBRI | 24.05.2018 10 / 36

slide-84
SLIDE 84

Compacted Binary Trees | A recurrence relation

A recurrence for compacted binary trees

Counting formula

Let n, p ∈ N, then γn+1,p =

n

  • i=0

γi,pγn−i,p+i, for n ≥ 1, Helps us to efficiently compute cn Asymptotic analysis failed (so far) One reason: asymptotically every summand matters Summands possess 3 (!) dependencies on i

Michael Wallner | LaBRI | 24.05.2018 11 / 36

slide-85
SLIDE 85

Compacted Binary Trees | A recurrence relation

A recurrence for compacted binary trees

Counting formula

Let n, p ∈ N, then γn+1,p =

n

  • i=0

γi,pγn−i,p+i, for n ≥ 1, γ0,p = p + 1, Helps us to efficiently compute cn Asymptotic analysis failed (so far) One reason: asymptotically every summand matters Summands possess 3 (!) dependencies on i

Michael Wallner | LaBRI | 24.05.2018 11 / 36

slide-86
SLIDE 86

Compacted Binary Trees | A recurrence relation

A recurrence for compacted binary trees

Counting formula

Let n, p ∈ N, then γn+1,p =

n

  • i=0

γi,pγn−i,p+i, for n ≥ 1, γ0,p = p + 1, γ1,p = p2 + p + 1. Helps us to efficiently compute cn Asymptotic analysis failed (so far) One reason: asymptotically every summand matters Summands possess 3 (!) dependencies on i

Michael Wallner | LaBRI | 24.05.2018 11 / 36

slide-87
SLIDE 87

Compacted Binary Trees | A recurrence relation

A recurrence for compacted binary trees

Counting formula

Let n, p ∈ N, then γn+1,p =

n

  • i=0

γi,pγn−i,p+i, for n ≥ 1, γ0,p = p + 1, γ1,p = p2 + p + 1. We are interested in cn = γn,0. Helps us to efficiently compute cn Asymptotic analysis failed (so far) One reason: asymptotically every summand matters Summands possess 3 (!) dependencies on i

Michael Wallner | LaBRI | 24.05.2018 11 / 36

slide-88
SLIDE 88

Compacted Binary Trees | A recurrence relation

A recurrence for compacted binary trees

Counting formula

Let n, p ∈ N, then γn+1,p =

n

  • i=0

γi,pγn−i,p+i, for n ≥ 1, γ0,p = p + 1, γ1,p = p2 + p + 1. We are interested in cn = γn,0. Helps us to efficiently compute cn Asymptotic analysis failed (so far) One reason: asymptotically every summand matters Summands possess 3 (!) dependencies on i

Michael Wallner | LaBRI | 24.05.2018 11 / 36

slide-89
SLIDE 89

Compacted Binary Trees | A recurrence relation

A recurrence for compacted binary trees

Counting formula

Let n, p ∈ N, then γn+1,p =

n

  • i=0

γi,pγn−i,p+i, for n ≥ 1, γ0,p = p + 1, γ1,p = p2 + p + 1. We are interested in cn = γn,0. Helps us to efficiently compute cn Asymptotic analysis failed (so far) One reason: asymptotically every summand matters Summands possess 3 (!) dependencies on i

Michael Wallner | LaBRI | 24.05.2018 11 / 36

slide-90
SLIDE 90

Compacted Binary Trees | A recurrence relation

A recurrence for compacted binary trees

Counting formula

Let n, p ∈ N, then γn+1,p =

n

  • i=0

γi,pγn−i,p+i, for n ≥ 1, γ0,p = p + 1, γ1,p = p2 + p + 1. We are interested in cn = γn,0. Helps us to efficiently compute cn Asymptotic analysis failed (so far) One reason: asymptotically every summand matters Summands possess 3 (!) dependencies on i

Michael Wallner | LaBRI | 24.05.2018 11 / 36

slide-91
SLIDE 91

Compacted Binary Trees | A recurrence relation

Relaxed compacted binary trees

Drop the condition of uniqueness of the subtrees, i.e. cn ≤ rn.

Michael Wallner | LaBRI | 24.05.2018 12 / 36

slide-92
SLIDE 92

Compacted Binary Trees | A recurrence relation

Relaxed compacted binary trees

Drop the condition of uniqueness of the subtrees, i.e. cn ≤ rn.

1 2 2 4 4 4 6 6

Michael Wallner | LaBRI | 24.05.2018 12 / 36

slide-93
SLIDE 93

Compacted Binary Trees | A recurrence relation

Relaxed compacted binary trees

Drop the condition of uniqueness of the subtrees, i.e. cn ≤ rn.

1 2 2 4 4 4 6 6

In total, this spine corresponds to 1 · 3 · 4 · 42 · 62 = 6912 relaxed trees. (Recall, that the same spine corresponds to 4836 compacted trees.)

Michael Wallner | LaBRI | 24.05.2018 12 / 36

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SLIDE 94

Compacted Binary Trees | A recurrence relation

Relaxed compacted binary trees of size 3

Michael Wallner | LaBRI | 24.05.2018 13 / 36

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SLIDE 95

Compacted Binary Trees | A recurrence relation

Relaxed compacted binary trees of size 3

Michael Wallner | LaBRI | 24.05.2018 13 / 36

The relaxed tree of size 3 which is not a compacted tree

compacted tree binary tree relaxed tree Reason: subtrees not unique

slide-96
SLIDE 96

Compacted Binary Trees | A recurrence relation

A recurrence for relaxed compacted binary trees

Counting formula

Let n, p ∈ N, then δn+1,p =

n

  • i=0

δi,pδn−i,p+i, for n ≥ 0, δ0,p = p + 1, ✭✭✭✭✭✭✭ ✭ ❤❤❤❤❤❤❤ ❤ δ1,p = p2 + p + 1. We are interested in rn = δn,0.

Michael Wallner | LaBRI | 24.05.2018 14 / 36

slide-97
SLIDE 97

Compacted Binary Trees | A recurrence relation

A recurrence for relaxed compacted binary trees

Counting formula

Let n, p ∈ N, then δn+1,p =

n

  • i=0

δi,pδn−i,p+i, for n ≥ 0, δ0,p = p + 1, ✭✭✭✭✭✭✭ ✭ ❤❤❤❤❤❤❤ ❤ δ1,p = p2 + p + 1. We are interested in rn = δn,0. Recursion still too complicated.

Michael Wallner | LaBRI | 24.05.2018 14 / 36

slide-98
SLIDE 98

Compacted Binary Trees | A recurrence relation

A recurrence for relaxed compacted binary trees

Counting formula

Let n, p ∈ N, then δn+1,p =

n

  • i=0

δi,pδn−i,p+i, for n ≥ 0, δ0,p = p + 1, ✭✭✭✭✭✭✭ ✭ ❤❤❤❤❤❤❤ ❤ δ1,p = p2 + p + 1. We are interested in rn = δn,0. Recursion still too complicated.

Example (Relaxed binary trees)

size n = 0 n = 1 n = 2 n = 3 n = 4 n = 5 n = 6 cn 1 1 3 15 111 1119 14487 rn 1 1 3 16 127 1363 18628

Michael Wallner | LaBRI | 24.05.2018 14 / 36

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SLIDE 99

Compacted Binary Trees | Operations on trees

Operations on trees

Michael Wallner | LaBRI | 24.05.2018 15 / 36

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SLIDE 100

Compacted Binary Trees | Operations on trees

Bounded right height

We restrict to a subclass of relaxed binary trees: bounded right height.

Michael Wallner | LaBRI | 24.05.2018 16 / 36

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SLIDE 101

Compacted Binary Trees | Operations on trees

Bounded right height

We restrict to a subclass of relaxed binary trees: bounded right height.

Right height

The right height of a binary tree is the maximal number of right children on any path from the root to a leaf.

Michael Wallner | LaBRI | 24.05.2018 16 / 36

slide-102
SLIDE 102

Compacted Binary Trees | Operations on trees

Bounded right height

We restrict to a subclass of relaxed binary trees: bounded right height.

Right height

The right height of a binary tree is the maximal number of right children on any path from the root to a leaf.

Example

A binary tree with right height 2. Nodes of level 0 are colored in red, nodes of level 1 in blue, and the node of level 3 in green.

Michael Wallner | LaBRI | 24.05.2018 16 / 36

slide-103
SLIDE 103

Compacted Binary Trees | Operations on trees

Bounded right height

We restrict to a subclass of relaxed binary trees: bounded right height.

Right height

The right height of a binary tree is the maximal number of right children on any path from the root to a leaf.

Example

← → A binary tree with right height 2. Nodes of level 0 are colored in red, nodes of level 1 in blue, and the node of level 3 in green.

Michael Wallner | LaBRI | 24.05.2018 16 / 36

slide-104
SLIDE 104

Compacted Binary Trees | Operations on trees

Compacted trees of right height ≤ k

n

Figure: Right height ≤ 0.

Michael Wallner | LaBRI | 24.05.2018 17 / 36

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SLIDE 105

Compacted Binary Trees | Operations on trees

Compacted trees of right height ≤ k

n

Figure: Right height ≤ 0. Figure: Right height ≤ 1.

Michael Wallner | LaBRI | 24.05.2018 17 / 36

slide-106
SLIDE 106

Compacted Binary Trees | Operations on trees

Compacted trees of right height ≤ k

n

Figure: Right height ≤ 0. Figure: Right height ≤ 1. Figure: Right height ≤ 2.

Michael Wallner | LaBRI | 24.05.2018 17 / 36

slide-107
SLIDE 107

Compacted Binary Trees | Operations on trees

Compacted trees of right height ≤ k

n

Figure: Right height ≤ 0. Figure: Right height ≤ 1. Figure: Right height ≤ 2. Figure: Right height ≤ 3.

Michael Wallner | LaBRI | 24.05.2018 17 / 36

slide-108
SLIDE 108

Compacted Binary Trees | Operations on trees

Motivational outlook

Theorem

The number rk,n of relaxed trees with right height at most k is for n → ∞ asymptotically equivalent to rk,n ∼ γkn!

  • 4 cos
  • π

k + 3 2n n−k/2, where γk ∈ R \ {0} is independent of n.

Michael Wallner | LaBRI | 24.05.2018 18 / 36

slide-109
SLIDE 109

Compacted Binary Trees | Operations on trees

Motivational outlook

Theorem

The number rk,n of relaxed trees with right height at most k is for n → ∞ asymptotically equivalent to rk,n ∼ γkn!

  • 4 cos
  • π

k + 3 2n n−k/2, where γk ∈ R \ {0} is independent of n.

Theorem (Main result)

The number ck,n of compacted trees with right height at most k is asymptotically equal to ck,n ∼ κkn!

  • 4 cos
  • π

k + 3 2n n− k

2 − 1 k+3 −( 1 4 − 1 k+3) cos( π k+3) −2

, where κk ∈ R \ {0} is independent of n.

Michael Wallner | LaBRI | 24.05.2018 18 / 36

slide-110
SLIDE 110

Compacted Binary Trees | Operations on trees

Main idea: Exponential generating functions

Asymptotic growth: n!ρn ⇒ exponential generating functions (EGF) Upper bound guarantees positive radius of convergence Problem: unlabeled structures!

Michael Wallner | LaBRI | 24.05.2018 19 / 36

slide-111
SLIDE 111

Compacted Binary Trees | Operations on trees

Main idea: Exponential generating functions

Asymptotic growth: n!ρn ⇒ exponential generating functions (EGF) Upper bound guarantees positive radius of convergence Problem: unlabeled structures!

Michael Wallner | LaBRI | 24.05.2018 19 / 36

slide-112
SLIDE 112

Compacted Binary Trees | Operations on trees

Main idea: Exponential generating functions

Asymptotic growth: n!ρn ⇒ exponential generating functions (EGF) Upper bound guarantees positive radius of convergence Problem: unlabeled structures!

Michael Wallner | LaBRI | 24.05.2018 19 / 36

slide-113
SLIDE 113

Compacted Binary Trees | Operations on trees

Main idea: Exponential generating functions

Asymptotic growth: n!ρn ⇒ exponential generating functions (EGF) Upper bound guarantees positive radius of convergence Problem: unlabeled structures! Idea: derive symbolic method for compacted trees

Michael Wallner | LaBRI | 24.05.2018 19 / 36

slide-114
SLIDE 114

Compacted Binary Trees | Operations on trees

Main idea: Exponential generating functions

Asymptotic growth: n!ρn ⇒ exponential generating functions (EGF) Upper bound guarantees positive radius of convergence Problem: unlabeled structures! Idea: derive symbolic method for compacted trees Let T(z) =

n≥0 tn zn n! be an EGF of the class T .

Michael Wallner | LaBRI | 24.05.2018 19 / 36

slide-115
SLIDE 115

Compacted Binary Trees | Operations on trees

Main idea: Exponential generating functions

Asymptotic growth: n!ρn ⇒ exponential generating functions (EGF) Upper bound guarantees positive radius of convergence Problem: unlabeled structures! Idea: derive symbolic method for compacted trees Let T(z) =

n≥0 tn zn n! be an EGF of the class T .

T(z) → zT(z)

Append a new node with a pointer to the class T .

T

Michael Wallner | LaBRI | 24.05.2018 19 / 36

slide-116
SLIDE 116

Compacted Binary Trees | Operations on trees

Main idea: Exponential generating functions

Asymptotic growth: n!ρn ⇒ exponential generating functions (EGF) Upper bound guarantees positive radius of convergence Problem: unlabeled structures! Idea: derive symbolic method for compacted trees Let T(z) =

n≥0 tn zn n! be an EGF of the class T .

T(z) → zT(z)

Append a new node with a pointer to the class T .

T

Proof: k![zk]zT(z) = k

  • ·

tk−1

  • Michael Wallner | LaBRI | 24.05.2018

19 / 36

slide-117
SLIDE 117

Compacted Binary Trees | Operations on trees

Main idea: Exponential generating functions

Asymptotic growth: n!ρn ⇒ exponential generating functions (EGF) Upper bound guarantees positive radius of convergence Problem: unlabeled structures! Idea: derive symbolic method for compacted trees Let T(z) =

n≥0 tn zn n! be an EGF of the class T .

T(z) → zT(z)

Append a new node with a pointer to the class T .

T

Proof: k![zk]zT(z) = k

  • k possible

pointers

· tk−1

  • k−1 internal

nodes

Michael Wallner | LaBRI | 24.05.2018 19 / 36

slide-118
SLIDE 118

Compacted Binary Trees | Operations on trees

Construction of R0(z)

Let R0(z) =

n≥0 r0,n zn n! be the EGF of relaxed binary trees with bounded right

height ≤ 0.

n

Michael Wallner | LaBRI | 24.05.2018 20 / 36

slide-119
SLIDE 119

Compacted Binary Trees | Operations on trees

Construction of R0(z)

Let R0(z) =

n≥0 r0,n zn n! be the EGF of relaxed binary trees with bounded right

height ≤ 0.

n

R0 = {}

  • Tree of size 0

∪ {◦} × R0

  • append new root

and new pointer

Michael Wallner | LaBRI | 24.05.2018 20 / 36

slide-120
SLIDE 120

Compacted Binary Trees | Operations on trees

Construction of R0(z)

Let R0(z) =

n≥0 r0,n zn n! be the EGF of relaxed binary trees with bounded right

height ≤ 0.

n

R0 = {}

  • Tree of size 0

∪ {◦} × R0

  • append new root

and new pointer

R0(z) = 1 1 − z =

  • n≥0

n!zn n!

Michael Wallner | LaBRI | 24.05.2018 20 / 36

slide-121
SLIDE 121

Compacted Binary Trees | Operations on trees

Construction of R0(z)

Let R0(z) =

n≥0 r0,n zn n! be the EGF of relaxed binary trees with bounded right

height ≤ 0.

1 2 3 n-3 n-2 n-1 n

R0 = {}

  • Tree of size 0

∪ {◦} × R0

  • append new root

and new pointer

R0(z) = 1 1 − z =

  • n≥0

n!zn n!

Michael Wallner | LaBRI | 24.05.2018 20 / 36

slide-122
SLIDE 122

Compacted Binary Trees | Operations on trees

Further constructions

S : T(z) →

1 1−z T(z)

Append a (possibly empty) sequence at the root.

T S = ∪ T ∪ T ∪

Michael Wallner | LaBRI | 24.05.2018 21 / 36

slide-123
SLIDE 123

Compacted Binary Trees | Operations on trees

Further constructions

S : T(z) →

1 1−z T(z)

Append a (possibly empty) sequence at the root.

T S = ∪ T ∪ T ∪

D : T(z) → d

dz T(z)

Delete top node but preserve its pointers.

T Michael Wallner | LaBRI | 24.05.2018 21 / 36

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SLIDE 124

Compacted Binary Trees | Operations on trees

Further constructions

S : T(z) →

1 1−z T(z)

Append a (possibly empty) sequence at the root.

T S = ∪ T ∪ T ∪

D : T(z) → d

dz T(z)

Delete top node but preserve its pointers.

T

I : T(z) →

  • T(z)

Add top node without pointers.

T

Michael Wallner | LaBRI | 24.05.2018 21 / 36

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SLIDE 125

Compacted Binary Trees | Operations on trees

Further constructions

S : T(z) →

1 1−z T(z)

Append a (possibly empty) sequence at the root.

T S = ∪ T ∪ T ∪

D : T(z) → d

dz T(z)

Delete top node but preserve its pointers.

T

I : T(z) →

  • T(z)

Add top node without pointers.

T

P : T(z) → z d

dz T(z)

Add a new pointer to the top node.

T

Michael Wallner | LaBRI | 24.05.2018 21 / 36

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SLIDE 126

Compacted Binary Trees | Relaxed binary trees

Relaxed binary trees

Michael Wallner | LaBRI | 24.05.2018 22 / 36

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SLIDE 127

Compacted Binary Trees | Relaxed binary trees

Construction of R1(z)

Let R1(z) =

ℓ≥0 r1,n zn n! be the EGF of relaxed binary trees with bounded right

height ≤ 1.

Michael Wallner | LaBRI | 24.05.2018 23 / 36

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SLIDE 128

Compacted Binary Trees | Relaxed binary trees

Construction of R1(z)

Let R1(z) =

ℓ≥0 r1,n zn n! be the EGF of relaxed binary trees with bounded right

height ≤ 1.

Decomposition of R1(z)

R1(z) =

  • n≥0

R1,ℓ(z) where R1,ℓ(z) is the EGF for relaxed binary trees with exactly ℓ left-subtrees, i.e. ℓ left-edges from level 0 to level 1.

Michael Wallner | LaBRI | 24.05.2018 23 / 36

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SLIDE 129

Compacted Binary Trees | Relaxed binary trees

Construction of R1(z)

Let R1(z) =

ℓ≥0 r1,n zn n! be the EGF of relaxed binary trees with bounded right

height ≤ 1.

Decomposition of R1(z)

R1(z) =

  • n≥0

R1,ℓ(z) where R1,ℓ(z) is the EGF for relaxed binary trees with exactly ℓ left-subtrees, i.e. ℓ left-edges from level 0 to level 1. R1,0(z) = R0(z) = 1 1 − z R1,1(z) = ?

Michael Wallner | LaBRI | 24.05.2018 23 / 36

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SLIDE 130

Compacted Binary Trees | Relaxed binary trees

Construction of R1,1(z)

Michael Wallner | LaBRI | 24.05.2018 24 / 36

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SLIDE 131

Compacted Binary Trees | Relaxed binary trees

Construction of R1,1(z)

Symbolic specification 1 delete initial sequence

Michael Wallner | LaBRI | 24.05.2018 24 / 36

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SLIDE 132

Compacted Binary Trees | Relaxed binary trees

Construction of R1,1(z)

Symbolic specification 1 delete initial sequence 2 decompose

Michael Wallner | LaBRI | 24.05.2018 24 / 36

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SLIDE 133

Compacted Binary Trees | Relaxed binary trees

Construction of R1,1(z)

Symbolic specification 1 delete initial sequence 2 decompose 3 append and add pointer

Michael Wallner | LaBRI | 24.05.2018 24 / 36

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SLIDE 134

Compacted Binary Trees | Relaxed binary trees

Construction of R1,1(z)

Symbolic specification 1 delete initial sequence 2 decompose 3 append and add pointer 4 add initial sequence R1,1(z)

R1,1(z) = S

  • init.

seq.

  • I
  • lvl 0

node

  • S ◦ P

red pointer and seq.

  • zR1,0(z)
  • non empty
  • R1,1(z) =

1 1 − z

  • 1

1 − z z (zR1,0(z))′ dz

Michael Wallner | LaBRI | 24.05.2018 24 / 36

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SLIDE 135

Compacted Binary Trees | Relaxed binary trees

Construction of R1,ℓ(z)

Michael Wallner | LaBRI | 24.05.2018 25 / 36

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SLIDE 136

Compacted Binary Trees | Relaxed binary trees

Construction of R1,ℓ(z)

Observation

Same structure as for R1,1(z)

R1,ℓ−1

R1,ℓ(z) = 1 1 − z

  • 1

1 − z z (zR1,ℓ−1(z))′ dz, ℓ ≥ 1, R1,0(z) = R0(z) = 1 1 − z .

Michael Wallner | LaBRI | 24.05.2018 25 / 36

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SLIDE 137

Compacted Binary Trees | Relaxed binary trees

Construction of R1,ℓ(z)

Observation

Same structure as for R1,1(z)

R1,ℓ−1

R1,ℓ(z) = 1 1 − z

  • 1

1 − z z (zR1,ℓ−1(z))′ dz, ℓ ≥ 1, R1,0(z) = R0(z) = 1 1 − z . Recall that R1(z) =

ℓ≥0 R1,ℓ(z). Summing the previous equation (formally) for

ℓ ≥ 1 gives 1 − 2z 1 − z R′

1(z) −

1 1 − z R1(z) − ((1 − z)R1,0(z))′ = 0.

Michael Wallner | LaBRI | 24.05.2018 25 / 36

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SLIDE 138

Compacted Binary Trees | Relaxed binary trees

Closed form of R1(z)

1 − 2z 1 − z R′

1(z) −

1 1 − z R1(z) − ((1 − z)R1,0(z))′ = 0. We know that R1,0(z) =

1 1−z and get

(1 − 2z) R′

1(z) − R1(z) = 0,

with R1(0) = 1.

Michael Wallner | LaBRI | 24.05.2018 26 / 36

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SLIDE 139

Compacted Binary Trees | Relaxed binary trees

Closed form of R1(z)

1 − 2z 1 − z R′

1(z) −

1 1 − z R1(z) − ((1 − z)R1,0(z))′ = 0. We know that R1,0(z) =

1 1−z and get

(1 − 2z) R′

1(z) − R1(z) = 0,

with R1(0) = 1. This directly yields R1(z) = 1 √1 − 2z .

Michael Wallner | LaBRI | 24.05.2018 26 / 36

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SLIDE 140

Compacted Binary Trees | Relaxed binary trees

Closed form of R1(z)

1 − 2z 1 − z R′

1(z) −

1 1 − z R1(z) − ((1 − z)R1,0(z))′ = 0. We know that R1,0(z) =

1 1−z and get

(1 − 2z) R′

1(z) − R1(z) = 0,

with R1(0) = 1. This directly yields R1(z) = 1 √1 − 2z . Therefore we get r1,n = n![zn]R1(z) = n! 2n 2n n

  • = (2n − 1)!!.

Michael Wallner | LaBRI | 24.05.2018 26 / 36

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SLIDE 141

Compacted Binary Trees | Relaxed binary trees

Closed form of R1(z)

1 − 2z 1 − z R′

1(z) −

1 1 − z R1(z) − ((1 − z)R1,0(z))′ = 0. We know that R1,0(z) =

1 1−z and get

(1 − 2z) R′

1(z) − R1(z) = 0,

with R1(0) = 1. This directly yields R1(z) = 1 √1 − 2z . Therefore we get r1,n = n![zn]R1(z) = n! 2n 2n n

  • = (2n − 1)!!.

Preprint (ArXiv:1706.07163): [W, 2017, “A bijection of plane increasing trees with relaxed binary trees of right height at most one”].

Michael Wallner | LaBRI | 24.05.2018 26 / 36

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SLIDE 142

Compacted Binary Trees | Relaxed binary trees

Bounded right height ≤ 2: R2(z)

Symbolic construction

  • 1 − 3z + z2

R′′

2 (z) + (2z − 3) R′ 2(z) = 0,

R2(0) = 1, R′

2(0) = 1,

then we get the closed form R′

2(z) =

1 1 − 3z + z2 , and the coefficients r2,n = n![zn]R2(z) = (n − 1)! √ 5

  • 3 +

√ 5 2 n −

  • 3 −

√ 5 2 n .

Michael Wallner | LaBRI | 24.05.2018 27 / 36

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SLIDE 143

Compacted Binary Trees | Relaxed binary trees

Bounded right height ≤ 3: R3(z)

Symbolic construction

  • 1 − 4z + 3z2

R′′′

3 (z) + (9z − 6) R′′ 3 (z) + 2R′ 3(z) = 0,

R3(0) = 1, R′

3(0) = 1, R′′ 3 (0) = 3

2, then we get the closed form R3(z) =

  • 3z − 2 +

√ 3 √ 1 − 4z + 3z2 √ 3 − 2 1/

√ 3

, and the asymptotics of the coefficients r3,n = n![zn]R3(z) = n! √ 6

  • 2 −

√ 3 1/

√ 3

3n n3/2√π

  • 1 + O

1 n

  • .

Michael Wallner | LaBRI | 24.05.2018 28 / 36

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SLIDE 144

Compacted Binary Trees | Relaxed binary trees

Differential operators

Theorem

Let (Lk)k≥0 be a family of differential operators given by L0 = (1 − z), L1 = (1 − 2z)D − 1, Lk = Lk−1 · D − Lk−2 · D2 · z, k ≥ 2. Then the exponential generating function Rk(z) for relaxed trees with right height ≤ k satisfies Lk · Rk = 0.

Michael Wallner | LaBRI | 24.05.2018 29 / 36

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SLIDE 145

Compacted Binary Trees | Relaxed binary trees

Differential operators

Theorem

Let (Lk)k≥0 be a family of differential operators given by L0 = (1 − z), L1 = (1 − 2z)D − 1, Lk = Lk−1 · D − Lk−2 · D2 · z, k ≥ 2. Then the exponential generating function Rk(z) for relaxed trees with right height ≤ k satisfies Lk · Rk = 0. (1 − 2z) d dz R1(z) − R1(z) = 0

Michael Wallner | LaBRI | 24.05.2018 29 / 36

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SLIDE 146

Compacted Binary Trees | Relaxed binary trees

Differential operators

Theorem

Let (Lk)k≥0 be a family of differential operators given by L0 = (1 − z), L1 = (1 − 2z)D − 1, Lk = Lk−1 · D − Lk−2 · D2 · z, k ≥ 2. Then the exponential generating function Rk(z) for relaxed trees with right height ≤ k satisfies Lk · Rk = 0. (1 − 2z) d dz R1(z) − R1(z) = 0 (z2 − 3z + 1) d2 dz2 R2(z) + (2z − 3) d dz R2(z) = 0

Michael Wallner | LaBRI | 24.05.2018 29 / 36

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SLIDE 147

Compacted Binary Trees | Relaxed binary trees

Differential operators

Theorem

Let (Lk)k≥0 be a family of differential operators given by L0 = (1 − z), L1 = (1 − 2z)D − 1, Lk = Lk−1 · D − Lk−2 · D2 · z, k ≥ 2. Then the exponential generating function Rk(z) for relaxed trees with right height ≤ k satisfies Lk · Rk = 0. (1 − 2z) d dz R1(z) − R1(z) = 0 (z2 − 3z + 1) d2 dz2 R2(z) + (2z − 3) d dz R2(z) = 0 (3z2 − 4z + 1) d3 dz3 R3(z) + (9z − 6) d2 dz2 R3(z) + 2 d dz R3(z) = 0

Michael Wallner | LaBRI | 24.05.2018 29 / 36

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SLIDE 148

Compacted Binary Trees | Relaxed binary trees

Asymptotics of relaxed trees with bounded right height

Theorem

The number rk,n of relaxed trees with right height at most k is for n → ∞ asymptotically equivalent to rk,n ∼ γkn!

  • 4 cos
  • π

k + 3 2n n−k/2, where γk ∈ R \ {0} is independent of n.

Michael Wallner | LaBRI | 24.05.2018 30 / 36

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SLIDE 149

Compacted Binary Trees | Relaxed binary trees

Sketch of proof

1 Let ℓk,i ∈ C[z] be such that

Lk = ℓk,k(z)Dk + ℓk,k−1(z)Dk−1 + . . . + ℓk,0(z). Find recurrences for ℓk,i(z) using Guess’n’Prove techniques.

2 Use singularity analysis directly on differential equation: 3 Exponential growth ρk: Roots of coefficient of leading polynomial ℓk,k(z) are

candidates.

4 ℓk,k(z) is a transformed Chebyshev polynomial of the second kind. Hence,

ρk = 1 4 cos

  • π

k+3

2 .

5 Subexponential growth: Use the indicial indicial polynomial derived from the

ℓk,i(z).

6 Find a basis of solutions for differential equation:

Only one is singular at ρk!

7 Prove that other coefficients ℓk,i(z) are nice.

Michael Wallner | LaBRI | 24.05.2018 31 / 36

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SLIDE 150

Compacted Binary Trees | Relaxed binary trees

Sketch of proof

1 Let ℓk,i ∈ C[z] be such that

Lk = ℓk,k(z)Dk + ℓk,k−1(z)Dk−1 + . . . + ℓk,0(z). Find recurrences for ℓk,i(z) using Guess’n’Prove techniques.

2 Use singularity analysis directly on differential equation: 3 Exponential growth ρk: Roots of coefficient of leading polynomial ℓk,k(z) are

candidates.

4 ℓk,k(z) is a transformed Chebyshev polynomial of the second kind. Hence,

ρk = 1 4 cos

  • π

k+3

2 .

5 Subexponential growth: Use the indicial indicial polynomial derived from the

ℓk,i(z).

6 Find a basis of solutions for differential equation:

Only one is singular at ρk!

7 Prove that other coefficients ℓk,i(z) are nice.

Michael Wallner | LaBRI | 24.05.2018 31 / 36

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SLIDE 151

Compacted Binary Trees | Relaxed binary trees

Sketch of proof

1 Let ℓk,i ∈ C[z] be such that

Lk = ℓk,k(z)Dk + ℓk,k−1(z)Dk−1 + . . . + ℓk,0(z). Find recurrences for ℓk,i(z) using Guess’n’Prove techniques.

2 Use singularity analysis directly on differential equation: 3 Exponential growth ρk: Roots of coefficient of leading polynomial ℓk,k(z) are

candidates.

4 ℓk,k(z) is a transformed Chebyshev polynomial of the second kind. Hence,

ρk = 1 4 cos

  • π

k+3

2 .

5 Subexponential growth: Use the indicial indicial polynomial derived from the

ℓk,i(z).

6 Find a basis of solutions for differential equation:

Only one is singular at ρk!

7 Prove that other coefficients ℓk,i(z) are nice.

Michael Wallner | LaBRI | 24.05.2018 31 / 36

slide-152
SLIDE 152

Compacted Binary Trees | Relaxed binary trees

Sketch of proof

1 Let ℓk,i ∈ C[z] be such that

Lk = ℓk,k(z)Dk + ℓk,k−1(z)Dk−1 + . . . + ℓk,0(z). Find recurrences for ℓk,i(z) using Guess’n’Prove techniques.

2 Use singularity analysis directly on differential equation: 3 Exponential growth ρk: Roots of coefficient of leading polynomial ℓk,k(z) are

candidates.

4 ℓk,k(z) is a transformed Chebyshev polynomial of the second kind. Hence,

ρk = 1 4 cos

  • π

k+3

2 .

5 Subexponential growth: Use the indicial indicial polynomial derived from the

ℓk,i(z).

6 Find a basis of solutions for differential equation:

Only one is singular at ρk!

7 Prove that other coefficients ℓk,i(z) are nice.

Michael Wallner | LaBRI | 24.05.2018 31 / 36

slide-153
SLIDE 153

Compacted Binary Trees | Relaxed binary trees

Sketch of proof

1 Let ℓk,i ∈ C[z] be such that

Lk = ℓk,k(z)Dk + ℓk,k−1(z)Dk−1 + . . . + ℓk,0(z). Find recurrences for ℓk,i(z) using Guess’n’Prove techniques.

2 Use singularity analysis directly on differential equation: 3 Exponential growth ρk: Roots of coefficient of leading polynomial ℓk,k(z) are

candidates.

4 ℓk,k(z) is a transformed Chebyshev polynomial of the second kind. Hence,

ρk = 1 4 cos

  • π

k+3

2 .

5 Subexponential growth: Use the indicial indicial polynomial derived from the

ℓk,i(z).

6 Find a basis of solutions for differential equation:

Only one is singular at ρk!

7 Prove that other coefficients ℓk,i(z) are nice.

Michael Wallner | LaBRI | 24.05.2018 31 / 36

slide-154
SLIDE 154

Compacted Binary Trees | Relaxed binary trees

Sketch of proof

1 Let ℓk,i ∈ C[z] be such that

Lk = ℓk,k(z)Dk + ℓk,k−1(z)Dk−1 + . . . + ℓk,0(z). Find recurrences for ℓk,i(z) using Guess’n’Prove techniques.

2 Use singularity analysis directly on differential equation: 3 Exponential growth ρk: Roots of coefficient of leading polynomial ℓk,k(z) are

candidates.

4 ℓk,k(z) is a transformed Chebyshev polynomial of the second kind. Hence,

ρk = 1 4 cos

  • π

k+3

2 .

5 Subexponential growth: Use the indicial indicial polynomial derived from the

ℓk,i(z).

6 Find a basis of solutions for differential equation:

Only one is singular at ρk!

7 Prove that other coefficients ℓk,i(z) are nice.

Michael Wallner | LaBRI | 24.05.2018 31 / 36

slide-155
SLIDE 155

Compacted Binary Trees | Relaxed binary trees

Sketch of proof

1 Let ℓk,i ∈ C[z] be such that

Lk = ℓk,k(z)Dk + ℓk,k−1(z)Dk−1 + . . . + ℓk,0(z). Find recurrences for ℓk,i(z) using Guess’n’Prove techniques.

2 Use singularity analysis directly on differential equation: 3 Exponential growth ρk: Roots of coefficient of leading polynomial ℓk,k(z) are

candidates.

4 ℓk,k(z) is a transformed Chebyshev polynomial of the second kind. Hence,

ρk = 1 4 cos

  • π

k+3

2 .

5 Subexponential growth: Use the indicial indicial polynomial derived from the

ℓk,i(z).

6 Find a basis of solutions for differential equation:

Only one is singular at ρk!

7 Prove that other coefficients ℓk,i(z) are nice.

Michael Wallner | LaBRI | 24.05.2018 31 / 36

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SLIDE 156

Compacted Binary Trees | Compacted binary trees

Compacted binary trees

Michael Wallner | LaBRI | 24.05.2018 32 / 36

slide-157
SLIDE 157

Compacted Binary Trees | Compacted binary trees

Compacted binary trees

Uniqueness of subtrees

Tk = − k2 + k + 1 (k + 1)2 k Tk Tk

Michael Wallner | LaBRI | 24.05.2018 33 / 36

slide-158
SLIDE 158

Compacted Binary Trees | Compacted binary trees

Compacted binary trees

Uniqueness of subtrees

Tk = − k2 + k + 1 (k + 1)2 k Tk Tk

Michael Wallner | LaBRI | 24.05.2018 33 / 36

slide-159
SLIDE 159

Compacted Binary Trees | Compacted binary trees

Compacted binary trees

Uniqueness of subtrees

Tk = − k2 + k + 1 (k + 1)2 k Tk Tk

Let (Mk)k≥0 be a family of differential operators such that the EGF Ck(z) for compacted binary trees with right height ≤ k satisfies Mk · Ck = 0.

Michael Wallner | LaBRI | 24.05.2018 33 / 36

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SLIDE 160

Compacted Binary Trees | Compacted binary trees

Compacted binary trees

Uniqueness of subtrees

Tk = − k2 + k + 1 (k + 1)2 k Tk Tk

Let (Mk)k≥0 be a family of differential operators such that the EGF Ck(z) for compacted binary trees with right height ≤ k satisfies Mk · Ck = 0. (1 − 2z) d2 dz2 C1(z) + (z − 3) d dz C1(z) = 0,

Michael Wallner | LaBRI | 24.05.2018 33 / 36

slide-161
SLIDE 161

Compacted Binary Trees | Compacted binary trees

Compacted binary trees

Uniqueness of subtrees

Tk = − k2 + k + 1 (k + 1)2 k Tk Tk

Let (Mk)k≥0 be a family of differential operators such that the EGF Ck(z) for compacted binary trees with right height ≤ k satisfies Mk · Ck = 0. (1 − 2z) d2 dz2 C1(z) + (z − 3) d dz C1(z) = 0, (z2 − 3z + 1) d3 dz3 C2(z) − (z2 − 6z + 6) d2 dz2 C2(z) − (2z − 3) d dz C2(z) = 0,

Michael Wallner | LaBRI | 24.05.2018 33 / 36

slide-162
SLIDE 162

Compacted Binary Trees | Compacted binary trees

Compacted binary trees

Uniqueness of subtrees

Tk = − k2 + k + 1 (k + 1)2 k Tk Tk

Let (Mk)k≥0 be a family of differential operators such that the EGF Ck(z) for compacted binary trees with right height ≤ k satisfies Mk · Ck = 0. (1 − 2z) d2 dz2 C1(z) + (z − 3) d dz C1(z) = 0, (z2 − 3z + 1) d3 dz3 C2(z) − (z2 − 6z + 6) d2 dz2 C2(z) − (2z − 3) d dz C2(z) = 0, (3z2 − 4z + 1) d4 dz4 C3(z) − (4z2 − 18z + 10) d3 dz3 C3(z) + · · · · · · + (z2 − 12z + 14) d2 dz2 C3(z) + (z − 3) d dz C3(z) = 0.

Michael Wallner | LaBRI | 24.05.2018 33 / 36

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SLIDE 163

Compacted Binary Trees | Compacted binary trees

Asymptotics of compacted trees with bounded right height

Theorem (Main result)

The number ck,n of compacted trees with right height at most k is asymptotically equal to ck,n ∼ κkn!

  • 4 cos
  • π

k + 3 2n n− k

2 − 1 k+3 −( 1 4 − 1 k+3) cos( π k+3) −2

, where κk ∈ R \ {0} is independent of n.

Michael Wallner | LaBRI | 24.05.2018 34 / 36

slide-164
SLIDE 164

Compacted Binary Trees | Compacted binary trees

Asymptotics of compacted trees with bounded right height

Theorem (Main result)

The number ck,n of compacted trees with right height at most k is asymptotically equal to ck,n ∼ κkn!

  • 4 cos
  • π

k + 3 2n n− k

2 − 1 k+3 −( 1 4 − 1 k+3) cos( π k+3) −2

, where κk ∈ R \ {0} is independent of n. Proof: We derived a symbolic method on exponential generating functions, leading to ordinary differential equations, and analyzed them by singularity analysis (recurrence relations on polynomial coefficients, indicial polynomial, transfer theorems).

Michael Wallner | LaBRI | 24.05.2018 34 / 36

slide-165
SLIDE 165

Compacted Binary Trees | Compacted binary trees

Asymptotics of compacted trees with bounded right height

Theorem (Main result)

The number ck,n of compacted trees with right height at most k is asymptotically equal to ck,n ∼ κkn!

  • 4 cos
  • π

k + 3 2n n− k

2 − 1 k+3 −( 1 4 − 1 k+3) cos( π k+3) −2

, where κk ∈ R \ {0} is independent of n. Proof: We derived a symbolic method on exponential generating functions, leading to ordinary differential equations, and analyzed them by singularity analysis (recurrence relations on polynomial coefficients, indicial polynomial, transfer theorems).

Michael Wallner | LaBRI | 24.05.2018 34 / 36

slide-166
SLIDE 166

Compacted Binary Trees | Compacted binary trees

Asymptotics of compacted trees with bounded right height

Theorem (Main result)

The number ck,n of compacted trees with right height at most k is asymptotically equal to ck,n ∼ κkn!

  • 4 cos
  • π

k + 3 2n n− k

2 − 1 k+3 −( 1 4 − 1 k+3) cos( π k+3) −2

, where κk ∈ R \ {0} is independent of n. Proof: We derived a symbolic method on exponential generating functions, leading to ordinary differential equations, and analyzed them by singularity analysis (recurrence relations on polynomial coefficients, indicial polynomial, transfer theorems).

Michael Wallner | LaBRI | 24.05.2018 34 / 36

slide-167
SLIDE 167

Compacted Binary Trees | Compacted binary trees

Asymptotics of compacted trees with bounded right height

Theorem (Main result)

The number ck,n of compacted trees with right height at most k is asymptotically equal to ck,n ∼ κkn!

  • 4 cos
  • π

k + 3 2n n− k

2 − 1 k+3 −( 1 4 − 1 k+3) cos( π k+3) −2

, where κk ∈ R \ {0} is independent of n. Proof: We derived a symbolic method on exponential generating functions, leading to ordinary differential equations, and analyzed them by singularity analysis (recurrence relations on polynomial coefficients, indicial polynomial, transfer theorems).

Michael Wallner | LaBRI | 24.05.2018 34 / 36

slide-168
SLIDE 168

Compacted Binary Trees | Compacted binary trees

Comparing compacted and relaxed trees

Compacted trees with right height at most k

ck,n ∼ κkn!

  • 4 cos
  • π

k + 3 2n n− k

2 − 1 k+3 −( 1 4 − 1 k+3) cos( π k+3) −2 Michael Wallner | LaBRI | 24.05.2018 35 / 36

slide-169
SLIDE 169

Compacted Binary Trees | Compacted binary trees

Comparing compacted and relaxed trees

Compacted trees with right height at most k

ck,n ∼ κkn!

  • 4 cos
  • π

k + 3 2n n− k

2 − 1 k+3 −( 1 4 − 1 k+3) cos( π k+3) −2

Relaxed trees with right height at most k

rk,n ∼ γkn!

  • 4 cos
  • π

k + 3 2n n−k/2

Michael Wallner | LaBRI | 24.05.2018 35 / 36

slide-170
SLIDE 170

Compacted Binary Trees | Compacted binary trees

Comparing compacted and relaxed trees

Compacted trees with right height at most k

ck,n ∼ κkn!

  • 4 cos
  • π

k + 3 2n n− k

2 − 1 k+3 −( 1 4 − 1 k+3) cos( π k+3) −2

Relaxed trees with right height at most k

rk,n ∼ γkn!

  • 4 cos
  • π

k + 3 2n n−k/2

Corollary (Proportion of compacted among relaxed trees)

ck,n rk,n

Michael Wallner | LaBRI | 24.05.2018 35 / 36

slide-171
SLIDE 171

Compacted Binary Trees | Compacted binary trees

Comparing compacted and relaxed trees

Compacted trees with right height at most k

ck,n ∼ κkn!

  • 4 cos
  • π

k + 3 2n n− k

2 − 1 k+3 −( 1 4 − 1 k+3) cos( π k+3) −2

Relaxed trees with right height at most k

rk,n ∼ γkn!

  • 4 cos
  • π

k + 3 2n n−k/2

Corollary (Proportion of compacted among relaxed trees)

ck,n rk,n ∼ κn

−

1 k+3 −( 1 4 − 1 k+3) 1 cos2( π k+3) Michael Wallner | LaBRI | 24.05.2018 35 / 36

slide-172
SLIDE 172

Compacted Binary Trees | Compacted binary trees

Comparing compacted and relaxed trees

Compacted trees with right height at most k

ck,n ∼ κkn!

  • 4 cos
  • π

k + 3 2n n− k

2 − 1 k+3 −( 1 4 − 1 k+3) cos( π k+3) −2

Relaxed trees with right height at most k

rk,n ∼ γkn!

  • 4 cos
  • π

k + 3 2n n−k/2

Corollary (Proportion of compacted among relaxed trees)

ck,n rk,n ∼ κn

−

1 k+3 −( 1 4 − 1 k+3) 1 cos2( π k+3) = o

  • n−1/4

.

Michael Wallner | LaBRI | 24.05.2018 35 / 36

slide-173
SLIDE 173

Compacted Binary Trees | Compacted binary trees

Next steps

Enumeration of compacted trees without height restrictions Different tree structures, like e.g. ternary trees Analyze shape parameters, like height, width, profile, . . .

Michael Wallner | LaBRI | 24.05.2018 36 / 36

slide-174
SLIDE 174

Compacted Binary Trees | Compacted binary trees

Next steps

Enumeration of compacted trees without height restrictions Different tree structures, like e.g. ternary trees Analyze shape parameters, like height, width, profile, . . .

Michael Wallner | LaBRI | 24.05.2018 36 / 36