Re-balancing the entire tree

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 entire tree:

p = inserted node; // p is for sure balanced... while ( p != root ) { p = p.parent; // traverse twards the root if ( p is unbalanced ) { /* ----------------------------------------------- Identify node x for the restructure operation ----------------------------------------------- */ z = p; y = the taller Child of z; x = the taller Child of y; p = restructure( x ); } }

Example:

Example execution of the Re-balance algorithm:

p = inserted node (46) while loop: p = p.parent (48) p (48) is balanced p = p.parent (50) p (50) is balanced p = p.parent (78) p (78) is unbalanced z = (78) y = (50) x = (48) p = rebalance( x (48) )

p = node (50) after rebalance returns.... p = p.parent (44) p (44) is now balanced: p is root, exit while loop

Re-balance algorithm in Java:

protected void rebalance(Position<Entry<K,V>> zPos) { if ( isInternal(zPos) ) setHeight(zPos); // update height of node while ( ! isRoot(zPos) ) { // traverse up the tree towards the root zPos = parent(zPos); setHeight(zPos); // update height of node if ( ! isBalanced(zPos) ) { /* -------------------- Find x -------------------- */ Position<Entry<K,V>> xPos = tallerChild(tallerChild(zPos)); zPos = restructure(xPos); // tri-node restructure setHeight(left(zPos)); // recompute heights setHeight(right(zPos)); setHeight(zPos); } } }

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

The insert method for the AVL tree

It is now very straightforward to write the insert method for an AVL tree:

/** * Inserts an item into the AVL Tree and returns the newly created * entry. */ public Entry<K,V> insert(K k, V v) throws InvalidKeyException { Entry<K,V> toReturn = super.insert(k, v); // insert into a binary search tree /* --------------------------------------------------------- Note: super.insert(k,v) update the "actionPos" variable actionPos = inserted node ---------------------------------------------------------- */ rebalance(actionPos); // rebalance up from the insertion position return toReturn; // Return a reference to the inserted node }

Note:

Example program

The directory containing the demo program is:
Example Program: (Demo the insert operation in AVL tree)
- The AVL Tree class file: click here
  - rebalance()
  - insert() (with rebalance)
- The BinarySearchTree Tree class file: click here
  - insert() (no rebalance)
  - restructure()
- Test program 1: click here
- Test program 1: click here
Output of Test Prog2: