Question

How do you delete a node in an AVL tree?

Answer 1

Deleting a node from an AVL tree involves two main steps: deleting the node as in a regular binary search tree deletion and then restoring the balance of the tree by performing rotations as needed. Here are the detailed steps for deleting a node from an AVL tree:

1. Perform Regular Binary Search Tree (BST) Deletion:

   • Find and delete the node as you would in a regular binary search tree. This involves finding the node to be deleted and handling three cases:
     1. If the node has no children, simply remove it.
     2. If the node has one child, replace the node with its child.
     3. If the node has two children, find the node's in-order successor (or predecessor), replace the node's value with the successor's (or predecessor's) value, and then delete the successor (or predecessor).

2. Update Heights:

   • After deleting the node, update the height of the ancestors of the deleted node.

3. Check Balance Factor:

   • Check the balance factor of each ancestor of the deleted node. If the balance factor becomes outside the acceptable range of -1, 0, or 1, the tree is unbalanced, and rotations are needed to restore balance.

4. Perform Rotations to Restore Balance:

   • Depending on the specific imbalance, perform the necessary rotations to restore balance. There are four cases similar to those in AVL tree insertion (LL, RR, LR, RL), and the rotations are applied to the lowest unbalanced ancestor of the deleted node.

Here's a more detailed explanation of the rotations for each case:

• Left-Left (LL) Imbalance:

  • Perform a right rotation on the unbalanced node.

• Right-Right (RR) Imbalance:

  • Perform a left rotation on the unbalanced node.

• Left-Right (LR) Imbalance:

  • Perform a left rotation on the left child, followed by a right rotation on the unbalanced node.

• Right-Left (RL) Imbalance:

  • Perform a right rotation on the right child, followed by a left rotation on the unbalanced node.

After performing the rotations, the balance factor of each node is updated, and the tree remains balanced.

It's important to note that during the deletion process, you may need to perform multiple rotations to restore balance, and the balancing operations may propagate up the tree to the root. The goal is to maintain the AVL tree's balanced property, ensuring that the balance factor of each node is within the acceptable range.