Introduction to Data Structure, Fall 2006 Slide- 1

California State University, Fresno

Introduction to Data Structure

Chapter 10

Ming LiDepartment of Computer Science

California State University, Fresno

Fall 2006

Introduction to Data Structure, Fall 2006 Slide- 2

California State University, Fresno

Binary Search Trees

A binary search tree (BST) is a binary tree with following properties:

1. Each node has a value. 2. A total order is defined on these values. 3. The left subtree of a node contains only values less than the node's

value. 4. The right subtree of a node contains only values greater than or equal

to the node's value.

Introduction to Data Structure, Fall 2006 Slide- 3

California State University, Fresno

Binary Search Trees

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Binary Search Tree 1

Introduction to Data Structure, Fall 2006 Slide- 4

California State University, Fresno

Binary Search Trees

B in ary Search T ree 2

5 3

3 0

5 9

6 2

5 0

B in ary Search T ree 3

Introduction to Data Structure, Fall 2006 Slide- 5

California State University, Fresno

Binary Search Tree Traversal – Inorder

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Introduction to Data Structure, Fall 2006 Slide- 6

California State University, Fresno

Current Node ActionRoot = 50 Compare item = 37 and 50

37 < 50, move to the left subtreeNode = 30 Compare item = 37 and 30

37 > 30, move to the right subtreeNode = 35 Compare item = 37 and 35

37 > 35, move to the right subtreeNode = 37 Compare item = 37 and 37. Item found.

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Binary Search Tree – Search

Introduction to Data Structure, Fall 2006 Slide- 7

California State University, Fresno

For a tree with root “r”, to search a node with a value of “target”

If(r->data == target)search is successful;

Else if(target < r->data)search in the left subtree with root “r->left”;

Elsesearch in the right subtree with root “r->right”

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Binary Search Tree – Search

Introduction to Data Structure, Fall 2006 Slide- 8

California State University, Fresno

bool BST_Search(struct tnode* root, int target){ if(root == NULL) return FALSE; if(r->data == target) return TRUE; else if(target < r->data)

return(BST_Search(root->left, target));else

return(BST_Search(root->right, target));}

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Binary Search Tree – Search

Introduction to Data Structure, Fall 2006 Slide- 9

California State University, Fresno

Current Node ActionRoot = 50 Compare item = 38 and 50

38 < 50, move to the left subtreeNode = 30 Compare item = 38 and 30

38 > 30, move to the right subtreeNode = 35 Compare item = 38 and 35

38 > 35, move to the right subtreeNode = 37 Compare item = 38 and 37

Got to right subtree. However, it is empty, so insert as right child!

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Binary Search Tree – Insertion

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Introduction to Data Structure, Fall 2006 Slide- 10

California State University, Fresno

Binary Search Tree Node – Insertion

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Insert 41 to the tree

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Introduction to Data Structure, Fall 2006 Slide- 11

California State University, Fresno

Binary Search Tree Node – Insertion

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Insert 31 to the tree

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How to build a BST, given the list of values?

Introduction to Data Structure, Fall 2006 Slide- 12

California State University, Fresno

Binary Search Tree – Insertion For a tree with root “r”, to insert a node with a value of “target”

If(r->data == target)node is already existed, return;

Else if (target < r->data)

if (r->left == NULL)insert as left child of “r”;

elseinsert in the left subtree with root “r->left”;

Else if (r->right == NULL)

insert as right child of “r”; else

insert in the right subtree with root “r->right”

Introduction to Data Structure, Fall 2006 Slide- 13

California State University, Fresno

Binary Search Tree – Insertion

For a tree with root “r”, to insert a node with a value of “target”

If(r == NULL)point r to the new node with value of target;

Else if (target < r->data)

insert in the left subtree with root “r->left”;

Elseinsert in the right subtree with root “r->right”

Introduction to Data Structure, Fall 2006 Slide- 14

California State University, Fresno

void InsertNode(struct node **node_ptr, struct node *newNode){

struct node *node = *node_ptr;

if (node == NULL)*node_ptr = newNode;

else if (newNode->value <= node->value)InsertNode(&node->left, newNode);

elseInsertNode(&node->right, newNode);

}

Binary Search Tree – Insertion

Introduction to Data Structure, Fall 2006 Slide- 15

California State University, Fresno

Binary Search Tree Traversal – Deletion

Challenge: How to maintain the order after deletion?

6 53 0

3 71 0

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1 5

\\ //

D elet e n o d e 2 5

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Bad Solution: 30 is out of place

Introduction to Data Structure, Fall 2006 Slide- 16

California State University, Fresno

Binary Search Tree Traversal – Deletion

Challenge: How to maintain the order after deletion?

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D elet e n o d e 2 5

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Good Solution

Introduction to Data Structure, Fall 2006 Slide- 17

California State University, Fresno

Binary Search Tree Traversal – Deletion

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Challenge: How to maintain the order after deletion?

Introduction to Data Structure, Fall 2006 Slide- 18

California State University, Fresno

Binary Search Tree Node – Deletion (1)

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Delete 15 from the tree

For leaf nodes, simply delete it.

Introduction to Data Structure, Fall 2006 Slide- 19

California State University, Fresno

Binary Search Tree Node – Deletion (2)

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Delete 60 from the tree

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For internal nodes with only one child, simply replace with that child.

Introduction to Data Structure, Fall 2006 Slide- 20

California State University, Fresno

Binary Search Tree Node – Deletion (3)

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Delete 10 from the tree

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For internal nodes with only one child, simply replace with that child.

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Introduction to Data Structure, Fall 2006 Slide- 21

California State University, Fresno

Binary Search Tree Node – Deletion (4)

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Delete 50 from the tree

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For internal nodes with two children, we can replace it with the previous in-order node.

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Introduction to Data Structure, Fall 2006 Slide- 22

California State University, Fresno

Binary Search Tree Node – Deletion (5)

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Delete 37 from to the tree

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For internal nodes with two children, we can also replace it with the next in-order node.

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Introduction to Data Structure, Fall 2006 Slide- 23

California State University, Fresno

Binary Search Tree Node – Deletion (6)

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Delete 25 from the tree

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For internal nodes with only one subtree, simply replace with the root of the subtree.

Introduction to Data Structure, Fall 2006 Slide- 24

California State University, Fresno

Binary Search Tree Node – Deletion (7)

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Delete 55 from the tree

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For internal nodes with only one subtree, we can replace it with the root of the subtree.

Introduction to Data Structure, Fall 2006 Slide- 25

California State University, Fresno

void DeleteNode(struct node*& node) {struct node*& temp = node;if (node->left == NULL) { //replace the node with the root of right subtree

node = node->right;delete temp;

} else if (node->right == NULL) { //left subtree is not empty but right subtree is empty // replace the node with the root of left subtree

node = node->left;delete temp;

} else {temp = node->left;while (temp->right != NULL) {

temp = temp->right;}node->value = temp->value;delete temp;

}}

Binary Search Tree – Deletion

Introduction to Data Structure, Fall 2006 Slide- 26

California State University, Fresno

1. How to find the next in-order node?

2. Check if a tree is a BST.

3. How to remove the duplicates. O(nlogn)

Binary Search Tree – Problems

Introduction to Data Structure, Fall 2006 Slide- 27

California State University, Fresno

Binary Search Tree – Next Node (1)

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Node 30

For internal nodes with right subtree, return the left-most leaf node of its right subtree.

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Introduction to Data Structure, Fall 2006 Slide- 28

California State University, Fresno

Binary Search Tree – Next Node (2)

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Node 50

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For internal nodes with right subtree, return the left-most leaf node of its right subtree.

Introduction to Data Structure, Fall 2006 Slide- 29

California State University, Fresno

Binary Search Tree – Next Node (3)

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Node 25

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For nodes without right subtree, if it is the left child of its parent, return its parent.

Introduction to Data Structure, Fall 2006 Slide- 30

California State University, Fresno

Binary Search Tree – Next Node (4)

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Node 31

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For nodes without right subtree, if it is the left child of its parent, return its parent.

Introduction to Data Structure, Fall 2006 Slide- 31

California State University, Fresno

Binary Search Tree – Next Node (5)

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Node 35

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For nodes without right subtree, if it is the right child of its parent, return the parent of its ancestor where the ancestor is the left child.

If there is no such ancestor, return NULL.

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Introduction to Data Structure, Fall 2006 Slide- 32

California State University, Fresno

Binary Search Tree – Next Node (6)

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Node 35

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For nodes without right subtree, if it is the right child of its parent, return the parent of its ancestor where the ancestor is the left child.

If there is no such ancestor, return NULL.

Introduction to Data Structure, Fall 2006 Slide- 33

California State University, Fresno

Binary Search Tree – Next Node (7)

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Node 62

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For nodes without right subtree, if it is the right child of its parent, return the parent of its ancestor where the ancestor is the left child.

If there is no such ancestor, return NULL.

NULL!

Introduction to Data Structure, Fall 2006 Slide- 34

California State University, Fresno

Binary Search Trees - Verification

int isBST(struct node* node){   if (node==NULL) return(true);

  if (node->left!=NULL && maxValue(node->left) > node->data)    return(false);

  if (node->right!=NULL && minValue(node->right) <= node->data)    return(false);

  if (!isBST(node->left) || !isBST(node->right))    return(false);   return(true); }

Introduction to Data Structure, Fall 2006 Slide- 35

California State University, Fresno

Using Binary Search Trees Application: Removing Duplicates

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