AVL Tree Basics Quiz

An AVL tree is a type of self-balancing binary search tree where each node must have a unique key value. This uniqueness is essential for maintaining the properties of the binary search tree, ensuring efficient search, insertion, and deletion operations. Therefore, multiple nodes with the same key value are not permitted in an AVL tree.

A left-left imbalance occurs when a node has a higher height on its left subtree, causing it to become unbalanced. To restore balance, a right rotation around the unbalanced node is performed, effectively shifting the left child up and the unbalanced node down, thus balancing the heights of the subtrees.

After a deletion in an AVL tree, the height may decrease if a node is removed. However, to maintain the AVL tree's balance property, rotations are performed as necessary. These rotations ensure that the tree remains balanced, preserving its logarithmic height and efficient search, insertion, and deletion operations.

AVL stands for Adelson-Velsky and Landis, named after the mathematicians who introduced this self-balancing binary search tree in 1962. The AVL tree maintains its balance by ensuring that the heights of the two child subtrees of any node differ by no more than one, which optimizes search, insertion, and deletion operations.

In an AVL tree, the balance factor is calculated as the height of the left subtree minus the height of the right subtree. This metric helps maintain the tree's balance, ensuring that the heights of the subtrees differ by no more than one, which is crucial for optimal search, insertion, and deletion operations.

In an AVL tree, the balance factor is defined as the height difference between the left and right subtrees of a node. A valid balance factor can only be -1, 0, or 1, ensuring that the tree remains balanced. This restriction allows efficient operations like insertion, deletion, and lookup while maintaining logarithmic height.

When a node has a balance factor of 2 with a right-heavy child, it indicates that the right subtree is taller. To restore balance, a left rotation is performed on the unbalanced node, effectively moving the right child up and the unbalanced node down, thereby rebalancing the tree structure.

An AVL tree maintains a strict balance through rotations during insertions and deletions, which guarantees that the height remains logarithmic relative to the number of nodes. This balance ensures efficient search operations, consistently achieving O(log n) time complexity, unlike regular binary search trees which can degrade to O(n) in the worst case when unbalanced.

In an AVL tree, a rotation is necessary when the balance factor of any node exceeds ±1, indicating that the tree has become unbalanced due to an insertion. This ensures that the tree maintains its height-balanced property, allowing for efficient search, insertion, and deletion operations.

A left-right rotation is a balancing technique used in binary search trees. It first applies a left rotation on the child node to adjust its position, followed by a right rotation on the parent node to maintain the overall structure and balance of the tree. This sequence effectively resolves imbalances caused by insertions.

AVL trees maintain a balance condition to ensure efficient operations. The maximum height of an AVL tree with n nodes is approximately 1.44 log(n+2), which reflects the logarithmic growth of the tree's height relative to the number of nodes. This ensures that operations like insertion, deletion, and lookup remain efficient, typically O(log n).

In an AVL tree, a node's balance factor is calculated as the difference between the heights of its left and right subtrees. When this factor becomes 2 or -2, it indicates a significant height imbalance, necessitating rotations to restore balance and maintain the AVL tree's properties.