Learning Objectives
By the end of this class, you should know...
- What is an AVL Tree?
- What is the definition of a balanced tree according to the AVL tree property?
- How do AVL trees maintain their balance?
- What is the pseudocode for AVL insert?
AVL Trees - An AVL tree (Georgy Adelson-Velsky and Evgenii Landis tree) is a self-balancing BST.
AVL Tree Property - Requires that the heights of the left and right children of every node differ by at most one.
|left.height - right.height| <= 1
- Maintains a balanced tree to achieve O(log n) time for ALL operations.
- Review: What is the Big-O runtime of BST operations in best, worst and average cases?
AVL Tree Insert - We need to insert nodes in such a way that the AVL tree property is satisfied.
- Thus, inside of our BST Nodes we are going to keep track of the height of the nodes.
struct Node
{
bstdata data; int height;
Node* left;
Node* right;
Node(): left(NULL), right(NULL), height(0){}
Node(bstdata newdata): left(NULL), right(NULL), data(newdata), height(0){}
}; - Call BST insert
- Rebalance the Tree
- To balance the tree we use what are called "rotations" to alter the location of certain nodes.
AVL Insert Pseudocode (Partial)
Node avlInsert(Node root, int value)
//BST style insertion
if (root == NULL)
return new Node(value)
else if(root.data >= value)
root->left = avlInsert(root->left, value)
else
root->right = avlInsert(root->right, value)
//value has now been inserted
//Is the resulting tree unbalanced?
int balance = root->left->height - root->right->height
if (balance > 1 || balance < -1)
//re-balance the tree using rotations
//to be filled in later in class
AVL Trees Maintain Balance by Readjusting the Tree -- i.e. doing what are called "Rotations" -- upon insertion and deletion.
  
Rotations 
- With rotations, we want the height of the tree to become shorter (tree looks less like a linked list)
- Important: These rotations maintain the order of the BST: T1 < A < T2 < B < T3.
Simple Examples
Image source. Right Rotation
Before Rotation After Rotation
  - Note that the upper case letters represent nodes in the tree, while the lower case letters represent possible subtrees.
- What happened to subtree c during the rotation?
Example Right Rotation
Right Rotation Pseudocode
Nodeptr Right-Rotate(Nodeptr root)
Nodeptr newRoot = root->left
root->left = newRoot->right
newRoot->right = root
root->height = maxHeight(root->left, root->right) + 1
newRoot->height = maxHeight(newRoot->left, newRoot->right) + 1
return newRoot
Left Rotation
Before Left Rotation After Left Rotation
Example Left Rotation 
Image source.
Double Rotations - Sometimes a single rotation is not enough to balance a tree.
- Consider the following example:
- Our reaction might be to right rotate it, like so:
- However, we are now no better off than when we started.
- In cases like these (zig-zag pattern), we need to use a double rotation.
- We will need to do either a left-right rotation or a right-left rotation, depending on the situation.
- If the left subree is "right heavy" (i.e. left.height - right.height < -1)
- rotate left then rotate right
- If the right subtree is "left heavy" (i.e. left.height - right.height > 1)
- rotate right then rotate left
- In the example above, we have a left heavy right subtree.
- Therefore, we will do a left-right rotation.
Step 1: Rotate left around the root.left, i.e. node A (to get a familar shape) Step 2: Rotate right around the root, i.e. B (to get a balanced tree)
Example of left-right rotation
Image source.- If we have the opposite situation, where the right subtree is left heavy:
- Then, we must perform a right-left rotation.
Step 1: Rotate right around root.right, i.e. A (to get a familar shape) Step 2: Rotate left around root, i.e. B (to get a balanced tree)
For More Information: Double Rotation Videos
AVL Insert Pseudocode (Full)
Nodeptr avlInsert(Nodeptr root, int value)
{
//BST style insertion
if (root == NULL)
return(newNode(value));
if (value < root.data)
root->left = avlInsert(root->left, value);
else
root->right = avlInsert(root->right, value);
//Check the balance
int balance = height(root->left) - height(root->right);
//Check for 4 cases
// Simple Right Rotation
if (balance > 1 && value < root->left->data)
return rightRotate(root);
// Simple Left Rotation
else if (balance < -1 && value > root->right->data)
return leftRotate(root);
// Left Right Case Double Rotation
else if (balance > 1 && value > root->left->value)
root->left = leftRotate(root->left);
return rightRotate(root);
// Right Left Case Double Rotation
else if (balance < -1 && value < root->right->data)
root->right = rightRotate(root->right);
return leftRotate(root);
//return the node pointer
return root;
For More Information: Video
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