Positions, Iterators, Tree Structures and Tree Traversals. Readings - Chapter 7.3, 7.4 and 8

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1 Positions, Iterators, Tree Structures an Reaings - Chapter 7.3, 7.4 an 8 1

2 Positional Lists q q q q A position acts as a marker or token within the broaer positional list. A position p is unaffecte by changes elsewhere in a list; the only way in which a position becomes invali is if an explicit comman is issue to elete it. A position instance is a simple object, supporting only the following metho: n P.getElement( ): Return the element store at position p. To provie for a general abstraction of a sequence of elements with the ability to ientify the location of an element, we efine a positional list ADT Goorich, Tamassia, Golwasser Lists an Iterators 2

3 Positional List ADT q Accessor methos: 2014 Goorich, Tamassia, Golwasser Lists an Iterators 3

4 Positional List ADT (2) q Upate methos: 2014 Goorich, Tamassia, Golwasser Lists an Iterators 4

5 Example q A sequence of Positional List operations: 2014 Goorich, Tamassia, Golwasser Lists an Iterators 5

6 Positional List Implementation q The most natural way to implement a positional list is with a oubly-linke list. prev next element noe heaer noes/positions trailer elements 2014 Goorich, Tamassia, Golwasser Lists an Iterators 6

7 Insertion q Insert a new noe, q, between p an its successor. p A B C p A B q C p X q A B X C 2014 Goorich, Tamassia, Golwasser Lists an Iterators 7

8 Deletion q Remove a noe, p, from a oubly-linke list. p A B C D A B C p D A B C 2014 Goorich, Tamassia, Golwasser Lists an Iterators 8

9 Iterators q An iterator is a software esign pattern that abstracts the process of scanning through a sequence of elements, one element at a time Goorich, Tamassia, Golwasser Lists an Iterators 9

10 The Iterable Interface q q q Java efines a parameterize interface, name Iterable, that inclues the following single metho: n iterator( ): Returns an iterator of the elements in the collection. An instance of a typical collection class in Java, such as an ArrayList, is iterable (but not itself an iterator); it prouces an iterator for its collection as the return value of the iterator( ) metho. Each call to iterator( ) returns a new iterator instance, thereby allowing multiple (even simultaneous) traversals of a collection Goorich, Tamassia, Golwasser Lists an Iterators 10

11 The for-each Loop q Java s Iterable class also plays a funamental role in support of the for-each loop syntax: is equivalent to: 2014 Goorich, Tamassia, Golwasser Lists an Iterators 11

12 Trees q Abstract moel of a hierarchical structure q A tree consists of noes with a parent-chil relation google images Tree Structures 12

13 Trees - Examples Computers R Us Sales Manufacturing R&D US International Laptops Desktops Europe Asia Canaa organization structure of a corporation 2014 Goorich, Tamassia, Golwasser Tree Structures 13

14 Trees - Examples (2) /user/rt/courses/ cs016/ cs252/ graes homeworks/ programs/ projects/ graes hw1 hw2 hw3 pr1 pr2 pr3 papers/ emos/ buylow sellhigh market Portion of a file system 2014 Goorich, Tamassia, Golwasser Tree Structures 14

15 Trees - Terminology q q q q q q q q A is the root noe B is parent of E an F A is ancestor of E an F E an F are escenants of A C is the sibling of B E an F are chilren of B E, I, J, K, G, H, an D are leaves A, B, C, an F are internal noes E B F I J K A G C H D Tree Structures 15

16 Trees - Terminology (2) q q q q q q q q A is the root noe B is parent of E an F A is ancestor of E an F E an F are escenants of A C is the sibling of B E an F are chilren of B E, I, J, K, G, H, an D are leaves A, B, C, an F are internal noes E q Subtree: tree consisting of noe an its escenants B F I J K A G C H D Tree Structures 16

17 Trees - Terminology (3) q The epth (level) of E is 2 q The height of the tree is 3 q The egree of noe F is 3 A level 0 B C D level 1 E F G H level 2 I J K level 3 Tree Structures 17

18 Binary Trees q An orere tree is one in which the chilren of each noe are orere B A C q Binary tree: orere tree with all noes having at most 2 chilren D H E I F G n left chil an right chil Tree Structures 18

19 Binary Trees q Recursive efinition of binary tree A n either a leaf or B C n an internal noe (the root) an one/two binary trees (left subtree an/or right subtree) D H E I F G Tree Structures 19

20 Example of Binary Trees - Arithmetic Expression Tree q Binary tree associate with an arithmetic expression n n internal noes: operators external noes: operans / ((((3 + 1) *3) / (9-5)+2))-((3*(7-4)) +6)) Tree Structures 20

21 Example of Binary Trees - Decision Tree q q Binary tree associate with a ecision process n n internal noes: questions with yes/no answer external noes: ecisions Example: ining ecision Yes Inian Foo No Sinhi Sweets North Inian Chinese Yes No Yes No Kartick Yo! China Dominos Tree Structures 21

22 Proper, Full, Complete Binary Trees q Proper/Full - Every noe has either zero or two chilren q Complete - every level except possibly the last is completely fille an all leaf noes are as left as possible. Tree Structures 22

23 Binary tree from a complete binary tree q A binary tree can be obtaine from appropriate complete binary tree by pruning. Tree Structures 23

24 Properties of a Binary Tree q Notations n n - number of noes n n E - number of leaves (external noes) n n I - number of internal noes n h - height of the tree q h+1 n 2 h+1-1 q 1 n E 2 h q h n I 2 h -1 q log(n+1) -1 h n-1 Tree Structures 24

25 Properties of Binary Trees (2) q n E n I + 1 q proof by inuction on n I n Tree with 1 noe has a leaf but no internal noe n Assume n E n I + 1 for tree with k-1 internal noes n A tree with k internal noes has k 1 internal noes in the left subtree an k-k 1-1 internal noes in the right subtree n By inuction n E (k 1 +1) + (k-k 1-1+1) = k+1 Tree Structures 25

26 Complete Binary Tree q level i has 2 i noes q In a tree of height h n leaves are at level h n n E = 2 h n n I = h-1 =2 h -1 n n I = n E - 1 n n = 2 h+1-1 q In a tree of n noes n n E is (n+1)/2 n h = log 2 (n E ) Tree Structures 26

27 Tree ADT q q q We use positions to abstract noes Generic methos: n n n n integer size() boolean isempty() Iterator iterator() Iterable positions() Accessor methos: n n n n position root() position parent(p) Iterable chilren(p) Integer numchilren(p) q Query methos: n boolean isinternal(p) n boolean isexternal(p) n boolean isroot(p) q Aitional upate methos may be efine by ata structures implementing the Tree ADT 2014 Goorich, Tamassia, Golwasser Tree Structures 27

28 Linke Structure for Binary Trees Noe Structure parent left element right Tree Structures 28

29 Linke Structure for Binary Trees Noe Structure parent B left element A B right D A D C E C E Coe Fragments 8.8, 8.9, 8.10, 8.11 Tree Structures 29

30 Linke Structure for General Trees Noe Structure parent element chilren Tree Structures 30

31 Linke Structure for General Trees Noe Structure parent element B chilren B A D F A D F C E C E Tree Structures 31

32 Computing Depth q p be a position within the tree T q calculate epth(p) public int epth(position<e> p) throws IllegalArgumentException {! } if (isroot(p))! return 0;! else! return 1 + epth(parent(p));! Tree Structures 32

33 Computing Height private int heightba() { // works, but quaratic worst-case time! }! Analysis: int h = 0;! for (Position<E> p : positions())! if (isexternal(p))// only consier leaf positions! h = Math.max(h, epth(p));! return h;! O(n + Σ pεl ( p +1)) is O(n 2 )! Tree Structures 33

34 ! Computing Height public int height(position<e> p) throws IllegalArgumentException {! }! int h = 0; for (Position<E> c : chilren(p))! h = Math.max(h, 1 + height(c));! return h;! Analysis! O(Σ p (c p +1))is O(n) // base case if p is external! Tree Structures 34

35 q Systematic way of visiting all noes in a tree in a specifie orer n preorer - processes each noe before processing its chilren n postorer - processes each noe after processing its chilren 35

36 Preorer Traversal Paper Title Abstract References

37 Preorer Traversal - Algorithm q Algorithm preorer(p) n perform the visit action for position p n for each chil c in chilren(p) o w preorer(c) q Example: n reaing a ocument from beginning to en 37

38 Postorer Traversal /user/rt/courses/ cs016/ cs252/ graes homeworks/ programs/ projects/ graes hw1 hw2 hw3 pr1 pr2 pr3 papers/ emos/ buylow sellhigh market 38

39 Postorer Traversal - Algorithm q Algorithm postorer(p) n for each chil c in chilren(p) o w postorer(c) n perform the visit action for position p q Example n u - isk usage comman in Unix 39

40 Traversals of Binary Trees q preorer(v) n visit(v) n preorer(v.leftchil()) n preorer(v.rightchil()) q postorer(v) n postorer(v.leftchil()) n postorer(v.rightchil()) n visit(v) 40

41 More Example of Traversals q Visit - printing the ata in the noe A q Preorer traversal n a b e h i c f g B C q Postorer traversal D E F G n h i e b f g c a H I 41

42 Application of Postorer Traversal q Evaluating Arithmetic Expressions /

43 Application of Postorer Traversal q Evaluating Arithmetic Expressions /

44 Application of Postorer Traversal q Evaluating Arithmetic Expressions /

45 Application of Postorer Traversal q Evaluating Arithmetic Expressions /

46 Application of Postorer Traversal q Evaluating Arithmetic Expressions /

47 Application of Postorer Traversal q Evaluating Arithmetic Expressions 2 /

48 Application of Postorer Traversal q Evaluating Arithmetic Expressions 2 /

49 Application of Postorer Traversal q Evaluating Arithmetic Expressions 2 /

50 Application of Postorer Traversal q Evaluating Arithmetic Expressions 2 /

51 Application of Postorer Traversal q Evaluating Arithmetic Expressions /

52 Inorer traversals q Visit the noe between the visit to the left an right subtree q Algorithm inorer(p) n If p has a left chil lc then w inorer(lc) n perform visit action for position p n If p has a right chil rc then w inorer(rc) 52

53 Example - Inorer Traversal q Inorer n b h e i a f c g A B C D E F G H I 53

54 Euler Tour Traversal q q q Generic traversal of a binary tree Inclues as special cases the preorer, postorer an inorer traversals Walk aroun the tree an visit each noe three times: n n n on the left (preorer) from below (inorer) on the right (postorer) + 2 L B R Goorich, Tamassia, Golwasser 54

55 Builing Tree from Preorer Traversal q Given the preorer traversal, can we uniquely etermine the binary tree? Preorer a b e h i c f g 55

56 Builing Tree from Postorer Traversal q Given the postorer traversal, can we uniquely etermine the binary tree? Postorer h i e b f g c a 56

57 Builing Tree from Pre- an In- Orer Traversals q Given the preorer an inorer traversals of a binary tree we can uniquely etermine the tree Preorer a b e h i c f g Inorer b h e i a f c g 57

58 Builing Tree from Pre- an In- Orer Traversals q Given the preorer an inorer traversals of a binary tree we can uniquely etermine the tree Preorer a b e h i c f g Inorer b h e i a f c g A 58

59 Builing Tree from Pre- an In- Orer Traversals q Given the preorer an inorer traversals of a binary tree we can uniquely etermine the tree Preorer a b e h i c f g Inorer b h e i a f c g A 59

60 Builing Tree from Pre- an In- Orer Traversals q Given the preorer an inorer traversals of a binary tree we can uniquely etermine the tree Preorer a b e h i c f g c f g Inorer b h e i a f c g f c g A 60

61 Builing Tree from Pre- an In- Orer Traversals q Given the preorer an inorer traversals of a binary tree we can uniquely etermine the tree Preorer a b e h i c f g c f g Inorer b h e i a f c g f c g A 61

62 Builing Tree from Pre- an In- Orer Traversals q Given the preorer an inorer traversals of a binary tree we can uniquely etermine the tree Preorer a b e h i c f g c f g Inorer b h e i a f c g f c g A C 62

63 Builing Tree from Pre- an In- Orer Traversals q Given the preorer an inorer traversals of a binary tree we can uniquely etermine the tree Preorer a b e h i c f g c f g Inorer b h e i a f c g f c g A C 63

64 Builing Tree from Pre- an In- Orer Traversals q Given the preorer an inorer traversals of a binary tree we can uniquely etermine the tree Preorer a b e h i c f g c f g Inorer b h e i a f c g f c g A C F G 64

65 Builing Tree from Pre- an In- Orer Traversals q Given the preorer an inorer traversals of a binary tree we can uniquely etermine the tree Preorer Inorer A a b e h i c f g b h e i a f c g b e h i c f g b h e i f c g C F G 65

66 Builing Tree from Pre- an In- Orer Traversals q Given the preorer an inorer traversals of a binary tree we can uniquely etermine the tree Preorer Inorer A a b e h i c f g b h e i a f c g b e h i c f g b h e i f c g C F G 66

67 Builing Tree from Pre- an In- Orer Traversals q Given the preorer an inorer traversals of a binary tree we can uniquely etermine the tree Preorer Inorer A a b e h i c f g b h e i a f c g b e h i c f g b h e i f c g B C F G 67

68 Builing Tree from Pre- an In- Orer Traversals q Given the preorer an inorer traversals of a binary tree we can uniquely etermine the tree Preorer Inorer A a b e h i c f g b h e i a f c g b e h i c f g b h e i f c g B C F G 68

69 Builing Tree from Pre- an In- Orer Traversals q Given the preorer an inorer traversals of a binary tree we can uniquely etermine the tree Preorer Inorer A a b e h i c f g b h e i a f c g b e h i c f g b h e i f c g B C e h i h e i F G 69

70 Builing Tree from Pre- an In- Orer Traversals q Given the preorer an inorer traversals of a binary tree we can uniquely etermine the tree Preorer Inorer A a b e h i c f g b h e i a f c g b e h i c f g b h e i f c g B C e h i h e i F G 70

71 Builing Tree from Pre- an In- Orer Traversals q Given the preorer an inorer traversals of a binary tree we can uniquely etermine the tree Preorer Inorer A a b e h i c f g b h e i a f c g b e h i c f g b h e i f c g B C e h i h e i E F G 71

72 Builing Tree from Pre- an In- Orer Traversals q Given the preorer an inorer traversals of a binary tree we can uniquely etermine the tree Preorer Inorer A a b e h i c f g b h e i a f c g b e h i c f g b h e i f c g B C e h i h e i E F G 72

73 Builing Tree from Pre- an In- Orer Traversals q Given the preorer an inorer traversals of a binary tree we can uniquely etermine the tree Preorer Inorer A a b e h i c f g b h e i a f c g b e h i c f g b h e i f c g B C e h i h e i E F G H I 73

74 Builing Tree from Pre- an In- Orer Traversals q Given the preorer an inorer traversals of a binary tree we can uniquely etermine the tree Preorer Inorer A a b e h i c f g b h e i a f c g b e h i c f g b h e i f c g B C e h i h e i E F G H I 74

75 Builing Tree from Pre- an In- Orer Traversals q Given the preorer an inorer traversals of a binary tree we can uniquely etermine the tree Preorer Inorer A a b e h i c f g b h e i a f c g b e h i c f g b h e i f c g B C e h i h e i E F G H I 75

76 Builing Tree from Pre- an In- Orer Traversals q Given the preorer an inorer traversals of a binary tree we can uniquely etermine the tree Preorer Inorer A a b e h i c f g b h e i a f c g b e h i c f g b h e i f c g B C e h i h e i D E F G H I 76

77 Builing Tree from Post- an In- Orer Traversals q Given the postorer an inorer traversals of a binary tree we can uniquely etermine the tree q The last noe visite in the postorer traversal is the root of the binary tree Postorer h i e b f g c a Inorer b h e i a f c g 77

78 Builing Tree from Post- an In- Orer Traversals q Given the postorer an inorer traversals of a binary tree we can uniquely etermine the tree q The last noe visite in the postorer traversal is the root of the binary tree Postorer h i e b f g c a Inorer b h e i a f c g A 78

79 Builing Tree from Post- an In- Orer Traversals q Given the postorer an inorer traversals of a binary tree we can uniquely etermine the tree q The last noe visite in the postorer traversal is the root of the binary tree Postorer h i e b f g c a Inorer b h e i a f c g A 79

80 Builing Tree from Post- an In- Orer Traversals q Given the postorer an inorer traversals of a binary tree we can uniquely etermine the tree q The last noe visite in the postorer traversal is the root of the binary tree Postorer h i e b f g c a f g c Inorer b h e i a f c g f c g A 80

81 Builing Tree from Post- an In- Orer Traversals q Given the postorer an inorer traversals of a binary tree we can uniquely etermine the tree q The last noe visite in the postorer traversal is the root of the binary tree Postorer Inorer A h i e b f g c a b h e i a f c g f g c f c g C 81

82 Builing Tree from Post- an In- Orer Traversals q Given the postorer an inorer traversals of a binary tree we can uniquely etermine the tree q The last noe visite in the postorer traversal is the root of the binary tree Postorer Inorer A h i e b f g c a b h e i a f c g f g c f c g C 82

83 Builing Tree from Post- an In- Orer Traversals q Given the postorer an inorer traversals of a binary tree we can uniquely etermine the tree q The last noe visite in the postorer traversal is the root of the binary tree Postorer Inorer A h i e b f g c a b h e i a f c g f g c f c g C F G 83

84 Builing Tree from Post- an In- Orer Traversals q Given the postorer an inorer traversals of a binary tree we can uniquely etermine the tree q The last noe visite in the postorer traversal is the root of the binary tree Postorer Inorer A h i e b f g c a b h e i a f c g h i e b f g c b h e i f c g C F G 84

85 Builing Tree from Post- an In- Orer Traversals q Given the postorer an inorer traversals of a binary tree we can uniquely etermine the tree q The last noe visite in the postorer traversal is the root of the binary tree Postorer Inorer A h i e b f g c a b h e i a f c g h i e b f g c b h e i f c g B C F G 85

86 Builing Tree from Post- an In- Orer Traversals q Given the postorer an inorer traversals of a binary tree we can uniquely etermine the tree q The last noe visite in the postorer traversal is the root of the binary tree Postorer h i e b f g c a f g c Inorer b h e i a f c g f c g A h i e b b h e i B C q an so on.. F G 86

87 Builing Tree from Pre- an Post- Orer Traversals q Given the pre an postorer traversal of a binary tree, can we uniquely reconstruct the tree? Preorer a b e h i c f g Postorer h i e b f g c a 87

88 Builing Tree from Pre- an Post- Orer Traversals q Given the pre an postorer traversal of a binary tree, can we uniquely reconstruct the tree? Preorer a b e h i c f g Postorer h i e b f g c a A 88

89 Builing Tree from Pre- an Post- Orer Traversals q Given the pre an postorer traversal of a binary tree, can we uniquely reconstruct the tree? Preorer a b e h i c f g Postorer h i e b f g c a A 89

90 Builing Tree from Pre- an Post- Orer Traversals q Given the pre an postorer traversal of a binary tree, can we uniquely reconstruct the tree? Preorer a b e h i c f g c f g Postorer h i e b f g c a f g c A C 90

91 Builing Tree from Pre- an Post- Orer Traversals q Given the pre an postorer traversal of a binary tree, can we uniquely reconstruct the tree? Preorer a b e h i c f g c f g Postorer h i e b f g c a f g c A C F G 91

92 Builing Tree from Pre- an Post- Orer Traversals q Given the pre an postorer traversal of a binary tree, can we uniquely reconstruct the tree? Preorer a b e h i c f g Postorer h i e b f g c a A c f g f g c b e h i h i e b C F G 92

93 Builing Tree from Pre- an Post- Orer Traversals q Given the pre an postorer traversal of a binary tree, can we uniquely reconstruct the tree? Preorer a b e h i c f g Postorer h i e b f g c a A c f g f g c b e h i h i e b C F G 93

94 Builing Tree from Pre- an Post- Orer Traversals q Given the pre an postorer traversal of a binary tree, can we uniquely reconstruct the tree? Preorer a b e h i c f g Postorer h i e b f g c a A c f g f g c b e h i h i e b B C F G 94

95 Builing Tree from Pre- an Post- Orer Traversals q Given the pre an postorer traversal of a binary tree, can we uniquely reconstruct the tree? Preorer a b e h i c f g Postorer h i e b f g c a A c f g f g c b e h i h i e b B C F G 95

96 Builing Tree from Pre- an Post- Orer Traversals q Given the pre an postorer traversal of a binary tree, can we uniquely reconstruct the tree? Preorer a b e h i c f g Postorer h i e b f g c a A c f g f g c b e h i h i e b B C F G 96

97 Builing Tree from Pre- an Post- Orer Traversals q Given the pre an postorer traversal of a binary tree, can we uniquely reconstruct the tree? Preorer a b e h i c f g Postorer h i e b f g c a A c f g f g c b e h i e h i h i e b h i e B C D E F G 97

98 Builing Tree from Pre- an Post- Orer Traversals q Given the pre an postorer traversal of a binary tree, can we uniquely reconstruct the tree? Preorer a b e h i c f g Postorer h i e b f g c a A c f g f g c b e h i e h i h i e b h i e B C D E F G 98

99 Builing Tree from Pre- an Post- Orer Traversals q Given the pre an postorer traversal of a binary tree, can we uniquely reconstruct the tree? Preorer a b e h i c f g Postorer h i e b f g c a A c f g f g c b e h i e h i h i e b h i e B C D E F G H I 99

100 Builing Tree from Pre- an Post- Orer Traversals q Given the pre an postorer traversal of a binary tree, can we uniquely reconstruct the tree? Preorer a b e i c Postorer i e b c a 100

101 Builing Tree from Pre- an Post- Orer Traversals q Given the pre an postorer traversal of a binary tree, can we uniquely reconstruct the tree? Preorer a b e i c Postorer i e b c a A B C E E I I I I 101

102 Builing Tree from Pre- an Post- Orer Traversals q Given the pre an postorer traversal of a binary tree, can we uniquely reconstruct the tree? Preorer a b e i c Postorer i e b c a A Only if the internal noes in a binary tree have exactly two chilren E B E C I I I I 102

103 Breath First Traversal q Visit all positions at epth, before visiting the positions at epth X X X O X O X X O X O X O X O X O O X O X O X X O X O

104 Let T be a binary tree with n noes an let D be the sum of the epths of all the external noes of T. Show that if T has the minimum number of external noes possible, then D is O(n) an if T has the maximum number of external noes possible, then D is O(nlogn) Problems 104

105 Let T be an orere tree with n >1. Is it possible that the preorer traversal of T visits the noes in the same orer as the postorer traversal of T? If so given an example, otherwise explain why. Is it possible that the preorer traversal of T visits the noes in the reverse orer of postorer traversal? Problems 105

106 Give the pseuocoe for a nonrecursive metho for performing an inorer traversal of a binary tree in linear time. Problems 106

107 Let T be a tree with n positions. Define the lowest common ancestor(lca) between two positions p an q as the lowest point in T that has both p an q as escenants. Given two positions p an q, escribe an efficient algorithm for fining the LCA of p an q. What is the running time of your algorithm? Problems 107