ECE 242. Data Structures

Transcription

1 ECE 242 Data Structures Lecture 21 Binary Search Trees Overview Problem: How do I represent data so that no data value is present more than once? Sets: three different implementations Ordered List Binary Search Tree HashTable Binary Search Tree Efficient implementation Nicely suited to OOP Code example is important: please review!

2 Overcoming Ordered List Inefficiency Ordered List requires a linear search through nodes Inefficient Can be implement a data structure to aid a binary search? Idea: Create a tree with a specific order! k <k >k Binary Search Tree Organize nodes so as to search easily Binary Search Tree (BST) the key of every node in the left subtree is smaller than the key of this node the key of every node in the right subtree is larger than the key of this node Make decisions as the tree is traversed

3 Examples Of BST These are BSTs This is not a BST Easier to find elements than using an ordered list Transform Linked List Into Binary Search Tree Suppose the numbers in a linked list are:, 4, 7, 15, 3, 18, (1) (2) (3) (4)

4 Transform Linked List Into Binary Search Tree Suppose the numbers in a linked list are:, 4, 7, 15, 3, 18, (5) (6) (7) 16 Application for BST Three fields in one database Student ID Student Name Score Use student ID as key Assume student ID is unique

5 BST Interface Basic operations: Insert/Remove/Search/Printall Others: height() public interface BSTinterface { public void insert(int id, String name, double score); public void remove(int id); public void search(int id); public void printall(); public int height(); } Search In BST It is very easy to locate an element in BST Use Recursive Implementation Suppose we want to locate node

6 Search In BST step 1: current root = Because 7<, search left subtree step 2: current root = 5 - Because 7 > 5, search right subtree - Step 3: current root = 7 target found SearchRecursive Method private BSTNode searchrecursive(bstnode r, int target) { if (r==null) return null; else if( target < r.id ) // search left subtree return searchrecursive(r.left, target); else if( target > r.id ) // search right subtree return searchrecursive(r.right, target); else // find the target return r; }

7 Search Method public void search(string id) { BSTNode tmp = searchrecursive( root, id); if( tmp==null ) System.out.println( "Student ID " + id + " is not in this database"); else { System.out.println( "student ID: " + tmp.id); System.out.println( "student Name: " + tmp.name); System.out.println( "His/Her score: " + tmp.score); } } Search Complexity In each level, only one comparison Cost: height of tree worst case: 6 worst case: 3

8 Traverse BST Pre-Order print current node first then print left subtree at last print right subtree In-Order print left subtree first then print current node at last print right subtree Post-Order print left subtree first then print right subtree at last print current node In-Order Traverse Prefer In-Order Traverse in BST printing results are well sorted by the key! private void printallrecursive(bstnode r) { } if( r==null ) return; if( r.left!=null ) printallrecursive( r.left ); System.out.println("student ID: " + r.id + " student Name: " + r.name + " His/Her score: " + r.score); if( r.right!=null ) printallrecursive( r.right ); How to implement Pre-Order Traverse? How to implement Post-Order Traverse?

9 Insert Method In BST Insert a new record in the database There are two cases case 1: database is empty case 2: database is not empty Case 1: database is empty BST tree is empty Create a new node as root New node has no children New node has no parent Case 2: database is not empty BST tree is not empty We need to determine the position to insert the new node. Similar to search part, we start from the root Insert Method: Case 2 Case 2: database is not empty Compare to the root node - If the key for new record is less than the key of root, new record should be inserted to the left subtree if left subtree is null, just insert the new node here - If the key for new record is greater than the key of root, new record should be inserted to the right subtree if right subtree is null, just insert the new node here Do it recursively

10 Example for Case 2 Suppose we want to insert 7 in the tree below 16 Example for Case 2: Step 1 current root is Because 7 <, 7 needs to be inserted to left subtree 16 16

11 Example for Case 2: Step 2 current root is 5 Because 7 > 5, 7 needs to be inserted to right subtree 5 s right subtree is null, So we find the position for Summary Binary search tree is an efficient way to store ordered data Search and insert methods can be implemented in a recursive fashion Can also be implemented with tail recursion In the best case a BST has a height of O(log n), where n is the number of elements in the tree In the worst case it has height O(n) Search, insert, and remove time is proportional to the height of the tree