Re-balancing the AVL tree after insertion

Recall that:

Only nodes that are on the path from the root to the newly inserted node can become unbalanced
(Because the height of these nodes are changed by the insertion of the new node):

Re-balancing the AVL tree after insertion:

p = inserted node; // p is for sure balanced... pp = p.parent; ppp = pp.parent; while ( pp != root (i.e., ppp != null) ) { if ( ppp is unbalanced ) { /* ----------------------------------------------- Identify node x for the restructure operation ----------------------------------------------- */ z = ppp; y = pp; x = p; p = restructure( x ); break; // Done } p = pp; // traverse twards the root pp = ppp; ppp = ppp.parent; }

Example:

Example execution of the Re-balance algorithm:

p = inserted node (46) pp = (48) ppp = (50) while loop: ppp (50) is balanced ### Move up 1 level in the tree ### p = pp (48) pp = ppp (50) ppp = ppp.parent (78) ppp (78) is unbalanced z = (78) y = (50) x = (48) p = rebalance( x (48) )

break... exit while loop

Re-balance algorithm in Java:

Discussed later along with the delete method
(The re-balance algorithm for the delete operation is more complicated)
(The re-balance algorithm for the delete operation can also be used to re-balance the AVL tree in the case of an insert operation)

Note on rebalance()

Note:

To re-balance an AVL tree in the insertion requires the re-balanceing of 1 (one) unbalanced node

Reason:

The re-balance() method restores the height of the unbalanced node position back to the previous height value
Using an old height value, the rest of the AVL tree is balanced

Therefore:

The re-balance() method for the insert() method does not need to re-balance all nodes on the path from the parent of the deleted node to the root node

Re-balancing for the delete operation:

We will see momentarily that in the delete operation, we need to re-balance every node on the path from the parent of the deleted node to the root node