1. Binary Search Trees

2. Binary search trees
A binary search tree is a binary tree where all elements in the left subtree are less than elements in the right subtree 2

3. Binary search trees
A binary search tree is a binary tree where all elements in the left subtree are less than elements in the right subtree
As we saw earlier, inorder traversal is
a b c d e f g h i j which is in sorted order
So this is a binary search tree 3

4. Binary search trees
A binary search tree is a binary tree whose inorder traversal is in sorted order
As we saw earlier, inorder traversal is
a b c d e f g h i j which is in sorted order
So this is a binary search tree
There are many more (binary) search trees that have inorder traversal a b c d e f g h i j 4

5. Binary search trees
A binary search tree is a binary tree whose inorder traversal is in sorted order
As we saw earlier, inorder traversal is
a b c d e f g h i j which is in sorted order
So this is a binary search tree
All entries are unique!
Typical use … representing a set 5

7. demo

8. Implementing search in a binary search tree
Can implement binary search in O(log n) time on average
Takes longer if the tree is badly balanced
For every node X, value in all nodes in left subtree of X are less than value in X, and value of all nodes in right subtree are greater than value in X
Algorithm is simple:
if x < node then search left subtree
if x > node then search right subtree
When the tree is balanced the path length to the leaves is log(n) 8

9. Find 17

10. Find 17

11. Find 17

12. Find 17

13. Find 17

14. Find 17
Fails to find 17

15. Insert New Node
Inserting a new node
This method works whether we are using the tree to implement a set, sequence etc. as long as it is an injective binary search tree.
Inserting a new node in general we must:
Add to bottom of tree at all times (add a leaf)
Keep the search tree property (left less than right) 15

17. Insert 40

18. Insert 40

19. Insert 40

20. Insert 40

21. Insert 40

22. Another illustration of insertion …

23. Adding a node with value x = 5. (Example)
Insert New Node
Adding a node with value x = 5. (Example) 6 3 2 4 23

24. Adding a node with value x = 5. (Example)
Insert New Node
Adding a node with value x = 5. (Example) 6 3 2 4
Compare 5 with node:
5 > 3 so go right 24

25. Adding a node with value x = 5. (Example)
Insert New Node
Adding a node with value x = 5. (Example) 6 3 2 4
Compare 5 with node:
5 < 6 so go left 25

26. Adding a node with value x = 5. (Example)
Insert New Node
Adding a node with value x = 5. (Example) 6 3 2 4
Compare 5 with node:
5 > 4 so go right 26

27. Adding a node with value x = 5. (Example)
Insert New Node
Adding a node with value x = 5. (Example) 3 2 6 4
Can’t go right as that is null
Insert node right of here, as a leaf 27

28. Adding a node with value x = 5. (Example)
Insert New Node
Adding a node with value x = 5. (Example) 3 2 6 4 5
Can’t go right as that is null
Insert node right of here, as a leaf 28

29. Why bother?

31. n=3, log(n+1)-1 = 1 ≤ height ≤ n-1

32. n=7, log(n+1)-1 = 2 ≤ height ≤ n-1

33. n=15, log(n+1)-1 = 3 ≤ height ≤ n-1

34. n=31, log(n+1)-1 = 4 ≤ height ≤ n-1

35. n=63, log(n+1)-1 = 5 ≤ height ≤ n-1

36. n=127, log(n+1)-1 = 6 ≤ height ≤ n-1

37. n=255, log(n+1)-1 = 7 ≤ height ≤ n-1

38. And this is crucial
height (h) log2(n+1)-1 ≤ h ≤ n-1
If we get it right we can access data in logarithmic time!

39. Log to the base 2 of …
2 = 1
4 = 2
8 = 3
16 = 4 …
1024 = 10

40. Log to the base 2 of …
2 = 1
4 = 2
8 = 3
16 = 4 …
1024 = 10
Brilliant!

41. java implementation

42. Node

43. Node

44. Node

45. Node
Default constructor

46. Node
parameterised constructor
note use of this.

47. Node
getters

48. Node
setters

49. Node
predicates

50. Node
For printing

51. The BSTree class
BinarySearchTree (BSTree)

52. Your mission, should you choose to accept it …

53. Implement insert, find and delete …

54. BSTree

55. BSTree
A binary search tree has a root
where the root is a node
It also has a size, where size is
the number of nodes in the tree

56. BSTree
Default constructor, an empty tree
with an empty root

57. BSTree
Get the root
Test if tree is empty
Get the size of the tree

58. BSTree
Insert string s into the tree

59. BSTree
Insert string s into the tree

60. BSTree
Insert string s into the subtree
rooted on the current node

61. BSTree
If s is in the left subtree …

62. BSTree
If s is in the left subtree and
we need to create a new node

63. BSTree
If s is in the right subtree …

64. BSTree
If s is in the right subtree and
we need to create a new node

65. finding a node

66. finding a node
BSTree

67. finding a node
BSTree
The string s is present in the tree
if the tree isn’t empty and we can
find a node that contains s

68. finding a node
BSTree
find string s in the subtree rooted
on the current node

69. finding a node
BSTree
If s is equal to the data in the current
node then return the current node
as a result.

70. finding a node
BSTree
If s is less than the data in the current
node and the current node has a left child
then try and find s in the subtree rooted
on the left child

71. finding a node
BSTree
If s is greater than the data in the current
node and the current node has a right child
then try and find s in the subtree rooted
on the right child

72. finding a node
BSTree
s is less than the current node and it has no left child
or s is greater than the current node and it has no
right child … deliver null!

73. Deletion of a node

74. Deletion of a node 16 11 30 7 13 20 40 18 25

75. Delete 7
Deletion of a node 16 11 30 7 13 20 40 18 25

76. Delete 7
Deletion of a node 16 11 30 7 13 20 40 18 25

77. Delete 7
Deletion of a node 16 11 30 13 20 40 18 25

78. Delete 11
Deletion of a node 16 11 30 13 20 40 18 25

79. Delete 11
Deletion of a node 16 11 30 13 20 40 18 25

80. Delete 11
Deletion of a node 16 11 30 13 20 40
Copy contents 18 25

81. Delete 11
Deletion of a node 16 13 30 13 20 40
Copy contents 18 25

82. Delete 11
Deletion of a node 16 13 30 13 20 40
Copy contents 18 25

83. Delete 11
Deletion of a node 16 13 30 13 20 40
Delete leaf 18 25

84. Delete 11
Deletion of a node 16 13 30 20 40 18 25

85. Delete 16
Deletion of a node 16 13 30 20 40 18 25

86. Delete 16
Deletion of a node 16 13 30 20 40 18 25

87. Deletion of a node Delete 16 16 13 30 20 40 18 25
Find the smallest node in the right subtree

88. Deletion of a node Delete 16 16 13 30 20 40 18 25
Find the smallest node in the right subtree

89. Deletion of a node Delete 16 16 13 30 20 40 18 25
Find the smallest node in the right subtree

90. Delete 16
Deletion of a node 16 13 30 20 40 18 25
Copy contents

91. Delete 16
Deletion of a node 18 13 30 20 40 18 25
Copy contents

92. Delete 16
Deletion of a node 18 13 30 20 40 18 25
Delete leaf

93. Delete 16
Deletion of a node 18 13 30 20 40 25
Delete leaf

94. Deletion of a node Delete 16 18 13 30 20 40 25
This process maintains the inorder property

95. There is a symmetric equivalent: find largest node in left branch …
Delete 16
Deletion of a node 18 13 30 20 40 25
This process maintains the inorder property
There is a symmetric equivalent: find largest node in left branch …

96. Deletion of a node Delete 16 16 13 30 20 40 18 25
Find the largest node in the left subtree

97. Deletion of a node Delete 16 16 13 30 20 40 18 25
Find the largest node in the left subtree

98. Delete 16
Deletion of a node 13 13 30 20 40 18 25
Copy contents

99. Delete 16
Deletion of a node 13 13 30 20 40 18 25
Delete leaf

100. Delete 16
Deletion of a node 13 30 20 40 18 25
Delete leaf

101. Deletion of a node Delete 16 40 13 20 30 25 18
Therefore we have 2 options resulting in two different trees, both valid 40 18 20 30 13 25

102. Well baby, is there an algorithm for this?
Deletion of a node
Well baby, is there an algorithm for this?

103. Deletion of a node

104. Deletion of a non-internal node (a node with less than 2 children)
Non-internal – less than 2 children
6 cases to consider

105. Deletion of a non-internal node
Non-internal – less than 2 children
Actually only 4 cases to consider

106. Deletion of a non-internal node
Assuming we have already found the node to delete

107. Deletion of a non-internal node
subsumed
Assuming we have already found the node to delete

108. Deletion of a non-internal node

109. Deletion of a non-internal node

110. Deletion of a non-internal node

111. Deletion of a non-internal node

112. Deletion of a non-internal node

113. Deletion of a non-internal node

114. Deletion of a non-internal node

115. Deletion of a non-internal node

117. Deletion of an internal node
Only 2 cases to consider

118. Deletion of an internal node
Internal – 2 children
Actually only 1 case to consider

119. Deletion of an internal node
get the node, call it w, to the right of this node

120. Deletion of an internal node
get the node, call it w, to the right of this node w

121. Deletion of an internal node
get the node, call it w, to the right of this node
get the “smallest” node, call it y, in the subtree rooted at w w

122. Deletion of an internal node
get the node, call it w, to the right of this node
get the “smallest” node, call it y, in the subtree rooted at w w y

123. Deletion of an internal node
get the node, call it w, to the right of this node
get the “smallest” node, call it y, in the subtree rooted at w w y

124. Deletion of an internal node
get the node, call it w, to the right of this node
get the “smallest” node, call it y, in the subtree rooted at w
replace what’s in node with what’s in y w y

125. Deletion of an internal node
get the node, call it w, to the right of this node
get the “smallest” node, call it y, in the subtree rooted at w
replace what’s in node with what’s in y y w y

126. Deletion of an internal node
get the node, call it w, to the right of this node
get the “smallest” node, call it y, in the subtree rooted at w
replace what’s in node with what’s in y
node y is not-internal, delete it (as before) y w y

127. Deletion of an internal node
get the node, call it w, to the right of this node
get the “smallest” node, call it y, in the subtree rooted at w
replace what’s in node with what’s in y
node y is not-internal, delete it (as before) y w y

128. Deletion of an internal node
get the node, call it w, to the right of this node
get the “smallest” node, call it y, in the subtree rooted at w
replace what’s in node with what’s in y
node y is not-internal, delete it (as before)

129. Groovey. But what if the node is the root?
Deletion of a node
Groovey. But what if the node is the root?

130. Deletion of the root
Only 4 cases to consider

131. Deletion of the root
Only 4 cases to consider … or is there?

132. Deletion of the root

133. Deletion of the root

134. Deletion of the root

135. Deletion of the root

136. Deletion of the root

137. Finding the node to delete
Deletion of a node
As before …

138. All of the code for deletion

139. Deletion of a node
All of the code

141. Is there a demo?