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AVL Tree

AVL Tree implementation in C++ using classes and templates.
This tree is a special case of augmented BST. AVL tree is a self-balancing tree, ie it prevents skewness while the insertion and deletion operation. Height of each subtree rooted at the current node is stored with the current node.
For each node:

height = 1 + max( height( left_child ), height( right_child ) )

Basic Operation

AVL tree always maintains a loose balance ie the heights of the subtrees on both sides of a node can differ by atmost one.
The balance in the tree heights is achieved by the following two operations:

Right Rotation

        y                              x
       / \      Right Rotation        / \
      x   C     -------------->      A   y
     / \                                / \
    A   B                              B   C

Here, x and y are nodes while A, B and C are AVL trees.
It can be observed that the right height increases by one while the left length may or may not decrease by one for an AVL tree.

Left Rotation

        x                              y
       / \       Left Rotation        / \
      A   y      ------------->      x   C
         / \                        / \
        B   C                      A   B

Here, x and y are nodes while A, B and C are AVL trees.
It can be observed that the left height increases by one while the right length may or may not decrease by one for an AVL tree.

Balancing

Following are the states of nodes and the way to balance the tree. Notice that the height difference at the root node is 2 (loose balance is maintained).
Here x and y are nodes while h,h+1 denote the height of tree.

Case I

        y                              x
       / \      Right Rotation        / \
      x   h     -------------->     h+1  y
     / \             on y               / \
   h+1 h/h+1                         h/h+1 h

Case II

        y                              y
       / \       Left Rotation        / \
      x   h      ------------->      z   h
     / \             on x           / \
    h  h/h+1                      h+1 h/h+1

Then,

        y                              z
       / \      Right Rotation        / \
      z   h     -------------->     h+1  y
     / \             on y               / \
   h+1 h/h+1                         h/h+1 h

Case III

        x                              y
       / \       Left Rotation        / \
      h   y      ------------->      x   h+1
         / \         on x           / \
     h/h+1 h+1                     h h/h+1

Case IV

        y                              y
       / \      Right Rotation        / \
      h   x     -------------->      h   z
         / \        on x                / \
      h/h+1 h                       h/h+1 h+1

Then,

        y                              z
       / \       Left Rotation        / \
      h   z      ------------->      y  h+1
         / \         on y           / \
      h/h+1 h+1                    h h/h+1