AVL Tree Self Balancing Rotations – Left Rotation (RR) explained

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.

For every node in an AVL Tree, the difference between the height of its left subtree and right subtree should not be more than 1. Whenever insertion makes a node unbalanced, AVL rotations are used to balance the tree again.
In this article, let’s look into the RR imbalance case and the Left Rotation used to fix it.

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 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.

What is the Left Rotation (RR) in AVL Tree?

The RR case happens when:

• a node becomes right-heavy;
• the newly inserted node lies in the right subtree of its right child.

This imbalance can be fixed using a single Left Rotation.
To do so

• Right child “R” of the unbalanced node “A” replaced the unbalanced node position and
• unbalanced node “A” becomes the left child of the replacing node “R”.
• If node “R” has any existing left child
  • then it will be moved as the right child of unbalanced node “A”.

Let’s take an example to understand it better.

As you can see in the above example node “47” becomes unbalanced after insertion of new node “65”.

Condition for Left Rotations

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 right child of the parent’s right child.

This is called an RR imbalance. To summarize it means

Unbalanced node  -> Right Heavy
Right child      -> Right Heavy

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 right child of the parent’s right child.
• if above condition holds good then
  • parent’s right child becomes the new parent.
  • current parent becomes the left child of new parent.
  • if new parent has any left child then it becomes the right child of current parent.
• return the new parent node.

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 ()
{}

C++ Code for Left Rotation

Let’s look into the sample code for this.

/* Left Rotate 
 *          x                      z
 *         / \                    / \
 *        y   z                  x   b
 *           / \     ---->      / \   \
 *          a   b              y   a   c
 *               \
 *  new entry --> 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;
}

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;
}

Time Complexity

The Right Rotation itself takes: because only a fixed number of pointer updates are required.
If we include insertion and finding the unbalanced node, AVL insertion remains: O(log n) because the height of an AVL Tree is O(log n).

Other AVL Tree Rotations

AVL Trees can become unbalanced in four different ways.

You can also check the other rotation cases:

• AVL Tree LL Case: Right Rotation Explained
• AVL Tree Left-Right (LR) Rotation
• AVL Tree Right-Left (RL) Rotation

For the complete balancing flow, you can also refer to your AVL Tree Basics, AVL Tree Insertion and AVL Tree Deletion articles.

Summary

The RR case occurs when:

Node         -> Right Heavy
Right Child  -> Right Heavy

The fix is:

Single Left Rotation

So the RR case can be remembered as:

RR Case -> Left Rotation