For the AVL Tree insert the value 18 as shown. What type of imbalance does it cause? Show the result after the balancing as well as the steps.
For the AVL Tree insert the value 18 as shown. What type of imbalance does it cause? Show the result after the balancing as well as the steps.
In an AVL tree, whenever an insertion or deletion operation is performed, the tree's balance is checked and, if necessary, restored through a series of rotations. These rotations are designed to maintain the balance factor of each node within the acceptable range of -1, 0, or 1. This self-balancing property makes AVL trees an excellent choice for applications that require fast search, insertion, and deletion operations.
Inserting the value 18 into the AVL tree results in a "left-right" imbalance at node 25. This type of imbalance occurs when a node's left subtree is right-heavy, and its right child becomes left-heavy due to an insertion.
A left-right imbalance typically occurs when the following sequence of events takes place during insertion or deletion in an AVL tree:
- An element is inserted into a left child's right subtree (causing the left child's balance factor to become -2).
- A subsequent element is inserted into the right child's left subtree (causing the right child's balance factor to become 2).
This sequence of operations results in a left-right imbalance. To correct a left-right imbalance, a double rotation is performed. There are two types of double rotations: left-right (LR) and right-left (RL) rotations, depending on whether the imbalance started on the left or right side.
The left-right rotation typically involves the following steps:
- A left rotation is performed on the left child's subtree (making it a left-heavy subtree).
- A right rotation is then performed on the original node.
The purpose of these rotations is to restore balance to the tree and ensure that the balance factors of all nodes are within the acceptable range of -1, 0, or 1.
By addressing left-right imbalances, AVL trees maintain their self-balancing property and continue to provide efficient search, insertion, and deletion operations with a logarithmic time complexity.
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