Content of slide Tree Binary tree Implementation Binary Search Tree BST Operations Traversal Insertion Deletion Types of BST Complexity in BST Applications of BST

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e-Infochips Institute of Training Research and Academics Limited
Binary Search Tree
Guided By:-
Mrs. Darshana Mistry
Presented By:-
Dharita Chokshi
Disha Raval
Himani Patel

2. Outlines
• Tree
• Binary tree Implementation
• Binary Search Tree
• BST Operations
• Traversal
• Insertion
• Deletion
• Types of BST
• Complexity in BST
• Applications of BST

3. Trees
Tree
• Each node can have 0 or more children
• A node can have at most one parent
Binary tree
• Tree with 0–2 children per node
• Also known as Decision Making Tree

4. Trees
Terminology
• Root  no parent
• Leaf  no child
• Interior  non-leaf
• Height  distance from root to leaf (H)

5. Why is h important?
The Tree operations like insert, delete, retrieve etc. are
typically expressed in terms of the height of the tree h.
So, it can be stated that the tree height h determines
running time!

6. Binary Tree Implementation
Class Node
{
int data; // Could be int, a class, etc
Node *left, *right; // null if empty
void insert ( int data ) { … }
void delete ( int data ) { … }
Node *find ( int data ) { … }
…
}

7. Binary Search Tree
Key property is value at node
• Smaller values in left subtree
• Larger values in right subtree
Example
X > Y
X < Z
Y
X
Z

8. Binary Search Tree
Examples
Binary
search trees
Not a binary
search tree
5
10
30
2 25 45
5
10
45
2 25 30
5
10
30
2
25
45

9. Difference between BT and BST
A binary tree is simply a tree in which each node can have at
most two children.
A binary search tree is a binary tree in which the nodes are
assigned values, with the following restrictions :
1. No duplicate values.
2. The left subtree of a node can only have values less than
the node
3. The right subtree of a node can only have values greater
than the node and recursively defined
4. The left subtree of a node is a binary search tree.
5. The right subtree of a node is a binary search tree.

10. Binary Tree Search Algorithm
TREE-SEARCH(x,k)
If x==NIL or k==x.key
return x
If k < x.key
return TREE-SEARCH(x.left,k)
else
return TREE-SEARCH(x.right,k)

11. BST Operations
Four basic BST operations
1
2
3
4
Traversal
Search
Insertion
Deletion

12. BST Traversal

13. Preorder Traversal
23 18 12 20 44 35 52
Root Left Right

14. Postorder Traversal
12 20 18 35 52 44 23
Left Right Root

15. Inorder Traversal
12 18 20 23 35 44 52
Produces a sequenced list
Left Root Right

16. 1 10
1
10
8 4 6 3 2 5
8
4
63
2 5
Inorder Traversal

17. 1 10
1
10
8 4 6 3 2 5
8
4
63
2 5
Inorder Traversal

18. 1 10
1
10
8 4 6 3 2 5
8
4
63
2 5
Inorder Traversal

19. 1 10
1
10
8 4 6 3 2 5
8
4
63
2 5
1
Inorder Traversal

20. 1 10
1
10
8 4 6 3 2 5
8
4
63
2 5
1
Inorder Traversal

21. 1 10
1
10
8 4 6 3 2 5
8
4
63
2 5
1
Inorder Traversal

22. 1 10
1
10
8 4 6 3 2 5
8
4
63
2 5
1
Inorder Traversal

23. 1 10
1
10
8 4 6 3 2 5
8
4
63
2 5
1
Inorder Traversal

24. 1 10
1
10
8 4 6 3 2 5
8
4
63
2 5
1
Inorder Traversal

25. 1 10
1
10
8 4 6 3 2 5
8
4
63
2 5
1
Inorder Traversal

26. 1 10
1
10
8 4 6 3 2 5
8
4
63
2 5
1 2
Inorder Traversal

27. 1 10
1
10
8 4 6 3 2 5
8
4
63
2 5
1 2
Inorder Traversal

28. 1 10
1
10
8 4 6 3 2 5
8
4
63
2 5
1 2
Inorder Traversal

29. 1 10
1
10
8 4 6 3 2 5
8
4
63
2 5
1 2 3
Inorder Traversal

30. 1 10
1
10
8 4 6 3 2 5
8
4
63
2 5
1 2 3
Inorder Traversal

31. 1 10
1
10
8 4 6 3 2 5
8
4
63
2 5
1 2 3
Inorder Traversal

32. 1 10
1
10
8 4 6 3 2 5
8
4
63
2 5
1 2 3 4
Inorder Traversal

33. 1 10
1
10
8 4 6 3 2 5
8
4
63
2 5
1 2 3 4
Inorder Traversal

34. 1 10
1
10
8 4 6 3 2 5
8
4
63
2 5
1 2 3 4
Inorder Traversal

35. 1 10
1
10
8 4 6 3 2 5
8
4
63
2 5
1 2 3 4
Inorder Traversal

36. 1 10
1
10
8 4 6 3 2 5
8
4
63
2 5
1 2 3 4 5
Inorder Traversal

37. 1 10
1
10
8 4 6 3 2 5
8
4
63
2 5
1 2 3 4 5
Inorder Traversal

38. 1 10
1
10
8 4 6 3 2 5
8
4
63
2 5
1 2 3 4 5
Inorder Traversal

39. 1 10
1
10
8 4 6 3 2 5
8
4
63
2 5
1 2 3 4 5 6
Inorder Traversal

40. 1 10
1
10
8 4 6 3 2 5
8
4
63
2 5
1 2 3 4 5 6
Inorder Traversal

41. 1 10
1
10
8 4 6 3 2 5
8
4
63
2 5
1 2 3 4 5 6
Inorder Traversal

42. 1 10
1
10
8 4 6 3 2 5
8
4
63
2 5
1 2 3 4 5 6
Inorder Traversal

43. 1 10
1
10
8 4 6 3 2 5
8
4
63
2 5
1 2 3 4 5 6 8
Inorder Traversal

44. 1 10
1
10
8 4 6 3 2 5
8
4
63
2 5
1 2 3 4 5 6 8
Inorder Traversal

45. 1 10
1
10
8 4 6 3 2 5
8
4
63
2 5
1 2 3 4 5 6 8
Inorder Traversal

46. 1 10
1
10
8 4 6 3 2 5
8
4
63
2 5
1 2 3 4 5 6 8 10
Inorder Traversal

47. 1 10
1
10
8 4 6 3 2 5
8
4
63
2 5
1 2 3 4 5 6 8 10
Inorder Traversal

48. Binary Tree Insertion

49. 1 10 8 4 6 3 2 5
Binary Tree Insertion

50. 1 10 8 4 6 3 2 5
Binary Tree Insertion

51. 1
1 10 8 4 6 3 2 5
Binary Tree Insertion

52. 1
1 10 8 4 6 3 2 5
Binary Tree Insertion

53. 1
1 10 8 4 6 3 2 5
Binary Tree Insertion

54. 1
1 10 8 4 6 3 2 5
Binary Tree Insertion

55. 1
10
1 10 8 4 6 3 2 5
Binary Tree Insertion

56. 1 10
1
10
8 4 6 3 2 5
Binary Tree Insertion

57. 1 10
1
10
8 4 6 3 2 5
Binary Tree Insertion

58. 1 10
1
10
8 4 6 3 2 5
Binary Tree Insertion

59. 1 10
1
10
8 4 6 3 2 5
Binary Tree Insertion

60. 1 10
1
10
8 4 6 3 2 5
8
Binary Tree Insertion

61. 1 10
1
10
8 4 6 3 2 5
8
Binary Tree Insertion

62. 1 10
1
10
8 4 6 3 2 5
8
Binary Tree Insertion

63. 1 10
1
10
8 4 6 3 2 5
8
Binary Tree Insertion

64. 1 10
1
10
8 4 6 3 2 5
8
Binary Tree Insertion

65. 1 10
1
10
8 4 6 3 2 5
8
Binary Tree Insertion

66. 1 10
1
10
8 4 6 3 2 5
8
4
Binary Tree Insertion

67. 1 10
1
10
8 4 6 3 2 5
8
4
Binary Tree Insertion

68. 1 10
1
10
8 4 6 3 2 5
8
4
Binary Tree Insertion

69. 1 10
1
10
8 4 6 3 2 5
8
4
Binary Tree Insertion

70. 1 10
1
10
8 4 6 3 2 5
8
4
Binary Tree Insertion

71. 1 10
1
10
8 4 6 3 2 5
8
4
Binary Tree Insertion

72. 1 10
1
10
8 4 6 3 2 5
8
4
Binary Tree Insertion

73. 1 10
1
10
8 4 6 3 2 5
8
4
6
Binary Tree Insertion

74. 1 10
1
10
8 4 6 3 2 5
8
4
6
Binary Tree Insertion

75. 1 10
1
10
8 4 6 3 2 5
8
4
6
Binary Tree Insertion

76. 1 10
1
10
8 4 6 3 2 5
8
4
6
Binary Tree Insertion

77. 1 10
1
10
8 4 6 3 2 5
8
4
6
Binary Tree Insertion

78. 1 10
1
10
8 4 6 3 2 5
8
4
6
Binary Tree Insertion

79. 1 10
1
10
8 4 6 3 2 5
8
4
6
Binary Tree Insertion

80. 1 10
1
10
8 4 6 3 2 5
8
4
63
Binary Tree Insertion

81. 1 10
1
10
8 4 6 3 2 5
8
4
63
Binary Tree Insertion

82. 1 10
1
10
8 4 6 3 2 5
8
4
63
Binary Tree Insertion

83. 1 10
1
10
8 4 6 3 2 5
8
4
63
Binary Tree Insertion

84. 1 10
1
10
8 4 6 3 2 5
8
4
63
Binary Tree Insertion

85. 1 10
1
10
8 4 6 3 2 5
8
4
63
Binary Tree Insertion

86. 1 10
1
10
8 4 6 3 2 5
8
4
63
Binary Tree Insertion

87. 1 10
1
10
8 4 6 3 2 5
8
4
63
Binary Tree Insertion

88. 1 10
1
10
8 4 6 3 2 5
8
4
63
2
Binary Tree Insertion

89. 1 10
1
10
8 4 6 3 2 5
8
4
63
2
Binary Tree Insertion

90. 1 10
1
10
8 4 6 3 2 5
8
4
63
2
Binary Tree Insertion

91. 1 10
1
10
8 4 6 3 2 5
8
4
63
2
Binary Tree Insertion

92. 1 10
1
10
8 4 6 3 2 5
8
4
63
2
Binary Tree Insertion

93. 1 10
1
10
8 4 6 3 2 5
8
4
63
2
Binary Tree Insertion

94. 1 10
1
10
8 4 6 3 2 5
8
4
63
2
Binary Tree Insertion

95. 1 10
1
10
8 4 6 3 2 5
8
4
63
2
Binary Tree Insertion

96. 1 10
1
10
8 4 6 3 2 5
8
4
63
2 5
Binary Tree Insertion

97. 1 10
1
10
8 4 6 3 2 5
8
4
63
2 5
Binary Tree Insertion

98. Binary Tree Deletion

99. 1 10
1
10
8 4 6 3 2 5
8
4
63
2 5
Binary Tree Deletion

100. 1 10
1
10
8 4 6 3 2 5
8
4
63
2 5
Binary Tree Deletion

101. 1 10
1
10
8 4 6 3 2 5
8
4
63
5
Binary Tree Deletion

102. 1 10
1
10
8 4 6 3 2 5
8
4
63
5
Binary Tree Deletion

103. 1 10
1
10
8 4 6 3 5
8
4
63
5
Binary Tree Deletion

104. 1 10
1
10
8 4 6 3 5
8
4
63
5
Binary Tree Deletion

105. 1 10
1
10
8 4 6 3 5
8
4
63
5
Binary Tree Deletion

106. 1 10
1
10
8 4 6 3 5
4
63
5
Binary Tree Deletion

107. 1 10
1
10
8 4 6 3 5
4
63
5
Binary Tree Deletion

108. 1 10
1
10
8 4 6 3 5
4
63
5
Binary Tree Deletion

109. 1 10
1
10
4 6 3 5
4
63
5
Binary Tree Deletion

110. 1 10
1
10
4 6 3 5
4
63
5
Binary Tree Deletion

111. 1 10
1
10
4 6 3 5
4
63
5
Binary Tree Deletion

112. 1 10
1
10
4 6 3 5
4
63
5
Binary Tree Deletion

113. 1 10
1
10
4 6 3 5
4
63
5
Binary Tree Deletion

114. 1 10
1
10
4 6 3 5
5
63
5
Binary Tree Deletion

115. 1 10
1
10
4 6 3 5
5
63
5
Binary Tree Deletion

116. 1 10
1
10
4 6 3 5
5
63
5
Binary Tree Deletion

117. 1 10
1
10
4 6 3 5
5
63
Binary Tree Deletion

118. 1 10
1
10
4 6 3 5
5
63
Binary Tree Deletion

119. 1 10
1
10
6 3 5
5
63
Binary Tree Deletion

120. Types of BST
Red-
Black
Tree

121. AVL Tree
AVL tree is a self-balancing Binary Search Tree (BST)
where the difference between heights of left and right
subtrees cannot be more than one for all nodes.

122. Red Black Tree
• Every node has a color
either red or black.
• Root of tree is always
black.
• There are no two
adjacent red nodes (A red
node cannot have a red
parent or red child).
• Every path from root to a
NULL node has same
number of black nodes.

123. Splay Tree
Automatically moves frequently accessed elements
nearer to the root for quick to access

124. Complexity in BST
Operation Average Worst Case Best Case
Search O(log n) O(n) O(1)
Insertion O(log n) O(n) O(1)
Deletion O(log n) O(n) O(1)

125. Applications of BST
• Used in many search applications where data is
constantly entering/leaving, such as the map and set
objects in many languages' libraries.
• Storing a set of names, and being able to lookup based
on a prefix of the name. (Used in internet routers.)
• Storing a path in a graph, and being able to reverse any
subsection of the path in O(log n) time. (Useful in
travelling salesman problems).
• Finding square root of given number
• allows you to do range searches efficiently.

126. Thank you