1. Binary Search Tree Neil Tang 01/28/2010
CS223 Advanced Data Structures and Algorithms

2. CS223 Advanced Data Structures and Algorithms
Class Overview
Definition
Operations: contains, findMin, findMax, insert, remove
Time complexity analysis
CS223 Advanced Data Structures and Algorithms

3. CS223 Advanced Data Structures and Algorithms
Definition
A Binary Search Tree is a special binary tree in which for each node X, the values of all the items in its left subtree are smaller than that in X and the values of all the items in its right subtree are larger than that in X.
CS223 Advanced Data Structures and Algorithms

4. CS223 Advanced Data Structures and Algorithms
Operations
contains: find out if a binary search tree contains a given item x.
CS223 Advanced Data Structures and Algorithms

5. CS223 Advanced Data Structures and Algorithms
Operations
findMin/findMax: find the smallest/largest item in a binary search tree.
CS223 Advanced Data Structures and Algorithms

6. CS223 Advanced Data Structures and Algorithms
Operations
insert: insert an item into a binary search tree. 14 2 9 5 12 18 19 17 13 15 2 9 5 12 18 19 17 14 15 13 2 9 5 12 18 19 17 14 15 13 14 14 2 9 5 12 18 19 17 13 15 2 9 5 12 18 19 17 13 15
CS223 Advanced Data Structures and Algorithms

7. CS223 Advanced Data Structures and Algorithms
Operations
CS223 Advanced Data Structures and Algorithms

8. CS223 Advanced Data Structures and Algorithms
Operations
remove: delete an item in a binary search tree. 15 15 5 3 10 6 7 23 16 18 20 12 5 16
Case 1: 3 12 20 18 23 10 13 6 7 15 5 3 10 6 7 23 16 18 20 13 12 15 5 3 10 6 7 23 18 20 13 12
Case 2: 15 5 3 10 6 7 23 16 18 20 13 12 15 5 3 6 23 16 18 20 13 12 10 7
Case 3:
CS223 Advanced Data Structures and Algorithms

9. CS223 Advanced Data Structures and Algorithms
Operations
CS223 Advanced Data Structures and Algorithms

10. Time Complexity Analysis
The time complexities of almost all operations on a binary search tree are O(d), where d is the depth (height) of the tree.
d = 3 = log(N+1)-1
= O(logN)
d = 7 = (N+1)/2 - 1
= O(N)
d = N-1 = O(N)
Theorem: The average height of a randomly built (insertion) binary search tree on N keys is (logN) . (pp.268 Cormen’s book)
CS223 Advanced Data Structures and Algorithms