1.2 operations of tree representations
TRANSCRIPT
.Binary Search TreesOperations and Implementation
Definition
Binary search tree is a binary tree that satisfies the following constraint.
•The key value of left child is smaller than its parent key
•the key value of right child is greater than its parent.
OPERATIONS
Insertion
Deletion
Search
Find Max
Find Min
Insertion
Steps
1.Locate the parent of the node to be inserted
2.create a binary tree node
3.insert the node (attach the node to the parent)
Do by yourself
Insert the values 13, 3, 4,12, 14,
10, 5, 1, 8, 2, 7, 9,11, 6, 18 in
that order, starting from an
empty tree.
Deletion Operation
Case 1: Node to be deleted has no children
•simply delete the node.Case 2: Node to be deleted has either left or right
empty subtree
•append nonempty subtree to its grand parent nodeCase 3: Node to be deleted has both left and right
subtree
•Find the inorder successor node of the node to be deleted
•Attach the right subtree of inorder successor node to its grand parent
•Replace the node to be deleted by its inorder successor
9
7
5
64 8 10
9
7
5
6 8 10
Case 1: removing a node with 2 EMPTY SUBTREES
parent
cursor
Removal in BST: Example
Removing 4replace the link in the parent with null
Case 2: removing a node with 2 SUBTREES
9
7
5
6 8 10
9
6
5
8 10
cursorcursor
- replace the node's value with the max value in the left subtree- delete the max node in the left subtree
44
Removing 7
Removal in BST: Example
What other elementcan be used as replacement?
9
7
5
6 8 10
9
7
5
6 8 10
cursor
cursor
parent
parent
the node has no left child:link the parent of the node to the right (non-empty) subtree
Case 3: removing a node with 1 EMPTY SUBTREE
Removal in BST: Example
9
7
5
8 10
9
7
5
8 10
cursor
cursor
parent
parent
the node has no right child:link the parent of the node to the left (non-empty) subtree
Case 4: removing a node with 1 EMPTY SUBTREE
Removing 5
4 4
Removal in BST: Example
Deletion - Example
20
13 34
2815 36
22 29
332521
3
51
Search Operation
1. Start at the root node
2. Branch either left or right repeatedly until the element is found
3. if the search ends with empty subtree, the element is not present in the tree.
Do the following operations
Delete 4,10, 27,13 from the tree
shown before and show the
step by step results.
IMPLEMENTATION
Definining the node
struct node
{
int key;
node *parent;
node *left;
node *right;
};
Insert operationvoid insert_key(int key)
{
//scan through the binary tree
node *prev=NULL, *curr=root;
while (curr != NULL)
{
prev = curr;
if (key < curr->key)
curr = curr->left;
else if (key > curr->key)
curr = curr->right;
else
{
printf("Data Already Exists");
return ;
}
}
//create a node and assign the key
node *newnode = (struct node *)malloc(sizeof(struct node));
newnode->key = key;
newnode->parent=NULL;
newnode->left = newnode->right = NULL;
//insert the node
if (prev==NULL)
root = newnode;
else
{
if (key < prev->key)
prev->left = newnode;
else
prev->right = newnode;
newnode->parent=prev;
}
}
In-order Traversal
void inorder_traversal(node *ptr)
{
if (ptr !=NULL)
{
inorder_traversal(ptr->left);
printf(" %d", ptr->key);
inorder_traversal(ptr->right);
}
}
Pre-order and Post-order traversalvoid preorder_traversal(node *ptr)
{
if (ptr !=NULL)
{
printf(" %d", ptr->key);
preorder_traversal(ptr->left);
preorder_traversal(ptr->right);
}
}
void postorder_traversal(node *ptr)
{
if (ptr !=NULL)
{
postorder_traversal(ptr->left);
postorder_traversal(ptr->right);
printf(" %d", ptr->key);
}
}
Delete Operationvoid delete_key(int key){
if(root==NULL){
printf("\nTree is empty");return;
}//z -m node to be deleted //y - succssor node - x successor subtreenode *z=search_key(key);node *x,*y;if(z==NULL)
printf("Key not found\n");else{
if(z->left==NULL || z->right==NULL)y=z;
elsey=Tree_Successor(z);
if(y->left!=NULL)x=y->left;
elsex=y->right;
x->parent=y->parent;
if(y->parent==NULL)
root=x;
else
{
if(y==y->parent->left)
y->parent->left=x;
else
y->parent->right=x;
}
if(y!=z)
z->key=y->key;
free(y);
}
}
if(x!=NULL)
Finding the inorder successornode* Tree_Successor(node *temp){
node *y;if(temp->right!=NULL){
temp=temp->right;while(temp->left!=NULL)
temp=temp->left;return temp;
}else{
y=temp->parent;while(y!=NULL && temp==y->right){
temp=y;y=y->parent;
}return y;
}}
Search Operation
node* search_key(int key){
node *prev=NULL, *curr=root;while (curr != NULL){
prev = curr;if (key < curr->key)
curr = curr->left;else if (key > curr->key)
curr = curr->right;else {
return curr;//key found}
}return NULL;//key not found
}
1. Start with this grid of 12 matchsticks, remove two of them so that there are only two squares left.
Answer
3.Move two matchsticks to make only four identical squares.
Answer
GUIDED READING
ASSESSMENT
1.Which traversal technique traverses the elements of binary search tree in ascending order.
A. pre-order traversal
B. post-order traversal
C. in-order traversal
D. converse post-order traversal.
Contd..
2. The height of the binary tree in the best and worst case is
A. n and log n respectively
B. log n and n respectively
C. log n and n/2 respectively
D. n/2 and log n respectively
Contd..
3. Create a binary search tree using the following operations: insert 3, insert 7, insert 8, insert 1, insert 5, insert 0, insert 4, insert 6, insert 9. Then, the left and right child of the inorder successor of node 5 iso0 and 6 respectivelyo4 and null respectivelyo3 and 9 respectivelyo4 and 6 respectively
Contd..
4.The data structure used in in-order traversal is
A. list
B. stack
C. queue
D. linked list
Contd..
5. The successor used in deletion operation of binary search tree is
A. inorder successor
B. preorder successor
C. postorder successor
D. converse preorder successor