by An AVL (Adelson-Velskii and Landis) Tree is a self balancing Binary Search Tree which has the following properties.
For any node “A”, the height of the left subtree of “A” and height of the right subtree of “A” differ by 1 at max.
In case of Binary search Trees worst case search complexity is O(n) in cases when the Binary Search Tree is skewed. In AVL tree, since heights of left and right subtree are balanced, hence search complexity improves to O(log n). Before going ahead have a look into AVL Tree Basics.
AVL Tree Class definition
Since AVL Tree is a Binary Search Tree also, hence in all code below we have used the same property to show how to use inheritance effectively.
In below code, we have define AVL class as a derived class of Binary Search Tree Base class. Please check here for more information related to Binary Search Tree implementations before going ahead.
Let’s have a look into basic C++ class definition for AVL Tree.
template <typename T>
class BinarySearchTree
{
public:
T m_data;
BinarySearchTree* m_left;
BinarySearchTree* m_right;
BinarySearchTree (T data);
virtual ~BinarySearchTree ();
};
template <typename T>
class AVL: public BinarySearchTree<T>
{
public:
int m_height;
AVL (T data);
~AVL ();
};
template <typename T>
AVL<T>::AVL (T data):BinarySearchTree<T> (data), m_height (0)
{}
template <typename T>
AVL<T>::~AVL ()
{}
AVL Tree Rotations
Whenever a new element is inserted into an AVL Tree, there is a chance of AVL tree becoming unbalanced. In order to rebalance AVL tree again to satisfy the height criteria, AVL tree rotations are performed.
In order to rebalance the tree, we start at the node inserted and travel up the tree, balancing each and every node of the tree if needed.
Right Left Rotations
In this case, single rotation operations are unable to rebalance the AVL Tree and hence double rotation is performed to rebalance the tree.
To do so first right rotation is being performed and then left rotation is performed. This operation is usually performed when right subtree of a node has height higher than the left subtree.
After these two operation unbalanced node becomes balanced.
Let’s take an example to understand it better.
- As you can see in the above example node “34” becomes unbalanced after insertion of new node “35”. In order to balance node “34”
- Right rotate operation is performed on right child “47” of the unbalance node “34”.
- After that Left rotate operation is performed on the unbalanced node “34”.
- Now the tree becomes balanced again.
Condition for Right Left Rotations
Right Left rotation operation is being applied only when following condition is met.
- After insertion of new node,
- if the parent node of the new node has higher right subtree than left subtree by >= 2
- new node is added as the left child of the parent’s right child.
Algorithm
- Insert the new node at it’s location using Binary Search Tree insertion algorithm
- Check the height_balance of the parent node.
- if the parent node of the new node has higher right subtree than left subtree by >= 2
- new node is added as the left child of the parent’s right child.
- if above condition holds good then
- perform right rotate operation on node’s right child.
- Perform left rotate operation on node.
- return the new parent node.
Let’s look into the sample code for this.
Before going ahead have a look into Left Rotate Operations and Right Rotate Operations.
/* Right Left Rotate
* x x z
* / \ / \ / \
* a y a z x y
* / \ -----> \ -----> / / \
* z b y a c b
* \ / \
* c c b
*
*/
template <typename T>
AVL<T>* right_left_rotate (AVL<T>* node_to_be_balanced)
{
node_to_be_balanced -> m_right = right_rotate (dynamic_cast<AVL<T>*> (node_to_be_balanced -> m_right));
return left_rotate (node_to_be_balanced);
}
/* Left Rotate
* x z
* / \ / \
* y z x b
* / \ ----> / \ \
* a b y a c
* \
* c
*/
template <typename T>
AVL<T>* left_rotate (AVL<T>* node_to_be_balanced)
{
AVL<T>* temp = dynamic_cast<AVL<T>*> (node_to_be_balanced -> m_right);
AVL<T>* left_child = dynamic_cast<AVL<T>*> (node_to_be_balanced -> m_right -> m_left);
temp -> m_left = node_to_be_balanced;
node_to_be_balanced -> m_right = left_child;
node_to_be_balanced -> m_height = height_of_node (node_to_be_balanced);
temp -> m_height = height_of_node (temp);
return temp;
}
/* Right Rotate
* x y
* / \ / \
* y z b x
* / \ ----> / / \
* b a c a z
* /
* c
*/
template <typename T>
AVL<T>* right_rotate (AVL<T>* node_to_be_balanced)
{
AVL<T>* temp = dynamic_cast<AVL<T>*> (node_to_be_balanced -> m_left);
AVL<T>* right_child = dynamic_cast<AVL<T>*> (temp -> m_right);
temp -> m_right = node_to_be_balanced;
node_to_be_balanced -> m_left = right_child;
node_to_be_balanced -> m_height = height_of_node (node_to_be_balanced);
temp -> m_height = height_of_node (temp);
return temp;
}
Height of the node function definition
template <typename T>
int height_of_node (BinarySearchTree<T>* root)
{
if (!root)
return 0;
int left = height_of_node (root -> m_left);
int right = height_of_node (root -> m_right);
if (left > right)
return 1 + left;
else
return 1 + right;
}
For checking out the usage of this function, check AVL Tree Insertion and AVL Tree Deletion operation.
