AVL Tree and Binary Tree

AVL Tree

AVL tree is a self-balancing binary search tree in which each node maintains an extra information called as balance factor whose value is either -1, 0 or +1

Balance Factor

Balance factor of a node in an AVL tree is the difference between the height of the left subtree and that of right subtree of that node.

Balance Factor = (Height of Left Subtree - Height of Right Subtree) or (Height of Right Subtree - Height of Left Subtree)

The self balancing property of an AVL tree is maintained by the balance factor. The value of balance factor should always be -1, 0 or +1.

An example of a balanced AVL tree is:

Operations on an AVL tree

Various operations that can be performed on an AVL tree are:

Rotating the subtrees in an AVL Tree

In rotation operation, the positions of the nodes of a subtree are interchanged.

There are two types of rotations:

Left Rotate

In left-rotation, the arrangement of the nodes on the right is transformed into the arrangements on the left node.

Algorithm

1. Let the initial tree be:
left-rotate

2. If y has a left subtree, assign x as the parent of the left subtree of y.
left-rotate

3. If the parent of x p is NULL, make y as the root of the tree.

4. Else if x is the left child of p, make y as the left child of p.

5. Else assign y as the right child of p.
left-rotate

6. Make y as the parent of x.
left-rotate

Right Rotate

In left-rotation, the arrangement of the nodes on the left is transformed into the arrangements on the right node.

1. Let the initial tree be:
right-rotate

2. If x has a right subtree, assign y as the parent of the right subtree of x.
right-rotate

3. If the parent of y is NULL, make x as the root of the tree.

4. Else if y is the right child of its parent p, make x as the right child of p.

5. Else assign x as the left child of p.
right-rotate

6. Make x as the parent of y.
right-rotate

Left-Right and Right-Left Rotate

In left-right rotation, the arrangements are first shifted to the left and then to the right.

1. Do left rotation on x-y.
left-right rotate

2. Do right rotation on y-z.
left-right rotate

In left-right rotation, the arrangements are first shifted to the right and then to the left.

1. Do right rotation on x-y.
right-left rotate

2. Do left rotation on z-y.
right-left rotate

Algorithm to insert a newNode

A newNode is always inserted as a leaf node with balance factor equal to 0.

1. Let the initial tree be:

Let the node to be inserted be:
new node

2. Go to the appropriate leaf node to insert a newNode using the following recursive steps.
Compare newKey with rootKey of current tree.

a. If newKey < rootKey, call insertion algorithm on left subtree of current node until the leaf node is reached.

b. Else if newKey > rootKey, call insertion algorithm on the right subtree of current node until the leaf node is reached.

c. Else, return leafNode.
avl tree insertion

3. Compare leafKey obtained from above steps with newKey:

a. If newKey < leafKey, make newNode as the leftChild of leafNode.

b. Else, make newNode as rightChild of leafNode.
avl tree insertion

4. Update balanceFactor of the nodes.
avl tree insertion

5. If the nodes are unbalanced, then rebalance the node.

a. If balanceFactor > 1, it means the height of the left subtree is greater than that of the right subtree. So, do right rotation or left-right rotation

a. If newNodeKey < leftChildKey do right rotation.

b. Else, do left-right rotation.
insertion in avl tree

b. If balanceFactor < -1, it means the height of the right subtree is greater than that of the left subtree. So, do right rotation or right-left rotation

a. If newNodeKey > rightChildKey do left rotation.

b. Else, do right-left rotation

6. The final tree is:
left-right insertion

Algorithm to Delete a node

A node is always deleted as a leaf node. After deleting a node, the balance factors of the nodes get changed. In order to rebalance the balance factor, suitable rotations are performed.

1. Locate nodeToBeDeleted (recursion is used to find nodeToBeDeleted in the code used below).
node to be deleted

2. There are three cases for deleting a node:

a. If nodeToBeDeleted is the leaf node (ie. does not have any child), then remove nodeToBeDeleted.

b. If nodeToBeDeleted has one child, then substitute the contents of nodeToBeDeleted with that of child. Remove the child.

c. If nodeToBeDeleted has two children, find the inorder successor w of nodeToBeDeleted (ie. node with minimum value of key in the right subtree).
node to be deleted

a. Substitute the contents of nodeToBeDeleted with that of w.
substitute the node to be deleted

b. Remove the leaf node w.
remove w

3. Update balanceFactor of the nodes.
update bf

4. Rebalance the tree if balance factor of any of the nodes is not equal to -1, 0 or 1.

a. If balanceFactor of currentNode > 1,

a. If balanceFactor of leftChild >= 0, do right rotation.
right-rotate

b. Else do left-right rotation.

b. If balanceFactor of currentNode < -1,

a. If balanceFactor of rightChild <= 0, do left rotation.

b. Else do right-left rotation.

5. The final tree is:
avl tree

Binary Tree

A complete binary tree is a binary tree in which all the levels are completely filled except possibly the lowest one, which is filled from the left.

A complete binary tree is just like a full binary tree, but with two major differences

1. All the leaf elements must lean towards the left.

2. The last leaf element might not have a right sibling i.e. a complete binary tree doesn't have to be a full binary tree.

Complete Binary Tree

Full Binary Tree vs Complete Binary Tree

Comparison between full binary tree and complete binary tree

How a Complete Binary Tree is Created?

1. Select the first element of the list to be the root node. (no. of elements on level-I: Complete binary tree creation

2. Put the second element as a left child of the root node and the third element as the right child. (no. of elements on level-II: 2) Complete binary tree creation

3. Put the next two elements as children of the left node of the second level. Again, put the next two elements as children of the right node of the second level (no. of elements on level-III: 4) elements).

4. Keep repeating until you reach the last element. Complete binary tree creation

Tugas :
#include <stdio.h>
#include <stdlib.h>

struct data
{
int x, height;
struct data *kiri;
struct data *kanan;
};

int max(int a, int b)
{
if(a < b) return b;
else return a;
}

int getHeight (struct data *root)
{
if(root==NULL) return 0;
return root->height;
}

int getBFactor (struct data *root)
{
if(root==NULL) return 0;
return getHeight(root->kiri) - getHeight(root->kanan);
}

struct data *kananRotate (struct data *y)
{
struct data *x = y->kiri;
struct data *B = x->kanan;

x->kanan = y;
y->kiri = B;

y->height = max(getHeight(y->kiri), getHeight(y->kanan)+1);
x->height = max(getHeight(x->kiri), getHeight(x->kanan)+1);

return x;
}

struct data *kiriRotate(struct data *x)
{
struct data *y = x->kanan;
struct data *B = y->kiri;

y->kiri = x;
x->kanan = B;

x->height = max(getHeight(x->kiri), getHeight(x->kanan)+1);
y->height = max(getHeight(y->kiri), getHeight(y->kanan)+1);

return y;
}

struct data *predecessor (struct data *root)
{
struct data *curr = root->kiri;

while(curr->kanan !=NULL)
curr = curr->kanan;

return curr;
}

struct data *successor (struct data *root)
{
struct data *curr = root->kanan;
while(curr->kiri !=NULL)
curr = curr->kiri;

return curr;
}

struct data *newNode (int x)
{
struct data *curr = (struct data*) malloc (sizeof(data));
curr->x = x;
curr->height = 1;
curr->kiri = NULL;
curr->kanan = NULL;

return curr;
}

struct data *insert (struct data *root, int x)
{
if(root == NULL) return newNode(x);
else if(x < root->x)
root->kiri = insert(root->kiri, x);
else
root->kanan = insert (root->kanan, x);

root->height = max(getHeight(root->kiri), getHeight(root->kanan))+1;
int bFactor = getBFactor(root);

if(bFactor > 1 && x < root->kiri->x)
return kananRotate(root);
if(bFactor <-1 && x > root->kanan->x)
return kiriRotate(root);

if(bFactor > 1 && x > root->kiri->x)
{
root->kiri = kiriRotate(root->kiri);
return kananRotate(root);
}

if(bFactor < -1 && x< root->kanan->x)
{
root->kanan = kananRotate(root->kanan);
return kiriRotate(root);
}

return root;
}

struct data *deleteValue (struct data *root, int x)
{
if(root == NULL) return NULL;
if(x < root->x)
root->kiri = deleteValue(root->kiri, x);
else if(x > root->x)
root->kanan = deleteValue(root->kanan, x);
else
{
if(root->kiri == NULL ||root->kanan == NULL)
{
struct data *temp = NULL;
if(root->kiri !=NULL)
temp = root->kiri;
else
temp = root->kanan;

if(temp == NULL)
{
temp = root;
root = NULL;
}
else
*root = *temp;

free(temp);
}
else
{

struct data *temp = predecessor(root);
root->x = temp->x;
}

}

if(root == NULL) return 0;
root->height = max(getHeight(root->kiri), getHeight(root->kanan))+1;

int bFactor = getBFactor(root);

if(bFactor > 1 && getBFactor(root->kiri)>=0)
return kananRotate(root);
if(bFactor <-1 && getBFactor(root->kanan)<=0)
return kiriRotate(root);

if(bFactor > 1 && getBFactor(root->kiri)<0)
{
root->kiri = kiriRotate(root->kiri);
return kananRotate(root);
}

if(bFactor < -1 && getBFactor(root->kanan)>0)
{
root->kanan = kananRotate(root->kanan);
return kiriRotate(root);
}

return root;
}

void printAll(struct data *root)
{
if(root==NULL) return;
printAll(root->kiri);
printf(" %d ", root->x);
printAll(root->kanan);
}

struct data *freeAll(struct data *root)
{
if(root ==NULL) return NULL;
root->kiri = freeAll(root->kiri);
root->kanan = freeAll(root->kanan);
free(root);

return NULL;
}

int main()
{
struct data *root = NULL;
root = insert(root,5);
root = insert(root,6);
root = insert(root,7);
root = insert(root,0);
root = insert(root,4);
root = insert(root,3);
root = insert(root,8);

printAll(root);
puts("");

root = deleteValue(root, 3);
root = deleteValue(root, 4);
root = deleteValue(root, 8);
printAll(root);

root = freeAll(root);

getchar();
return 0;
}

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Data Structures

AVL Tree and Binary Tree

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