AVL Tree

• AVL tree is a height balanced binary search tree.
• Balance factor: the difference between the height of right subtree and that of left subtree.
• For each node in an AVL tree, the height of its right subtree and left subtree differs at most 1.
• Theorem: the height of an AVL tree with N nodes is O(logN).

·  
· Proof: We prove that an AVL tree with height of h has at
·  least Omega(ch) nodes for some constant > 1.
· Base case: h = 1, AVL tree must have 1 node, so n(1) = 1.
·  h = 2, AVL tree must have at least 2 nodes,
·  so n(2) >= 2 >= 22/1
· Induction step: for h > 2,
·  n(h) >= 1 + n(h-1) + n(h-2)
·  > 2 * n(h-2)
·  >= 2 * 2(h-2)/2
·  >= 2h/2
·  >= sqrt(2)h
·  Therefore, if n(h) = N, h <= 2logN.

• Maintaining the balance of the AVL tree
• Insert a new node:
  Perform the normal BST insert while keep track of the last critical node;
  Update the balance factors of the affected nodes; the update only may propagate all the way to the root node;
  If a critical node is found, fix the possible unbalance using a single or a double rotation;
  The rotation shouldn't affect the height of the subtree, therefore, the rotation won't propagate upwards.
  The three rotation situations when inserting to the right subtree:

1. Single Rotation:

2. Double Rotation I:

3. Double Rotation II:

• Remove a node:
  Perform the normal BST removal while keep track of the last critical node;
  If a critical node is found, update the balance factors of the affected nodes;
  If unbalance happened, fix it using a single or a double rotation and update the balance factors of the affected nodes;
  Both the above actions may decrease the height of the subtree, therefore, the action may propagate to the parent nodes until hit the root node.
  The 5 remove situations when remove from the left subtree:

1. No rotation. No change in height. Done.

2. Single rotation. No change in height. Done.

3. No rotation. The height decrease by one. Continue up to parent node.

4. Single rotation. Decrease the height by one. Continue up to the parent node.

5. Double rotation. Decrease the height by one. Continue up to the parent node.