Complexity of Search Algorithm:

The complexity of a search algorithm refers to the efficiency with which the algorithm can locate a particular element in a data structure. In the context of binary search trees (BSTs), the complexity of the search algorithm depends on the height of the tree.

For a well-balanced BST:

• Average Case: Searching for an element in a balanced BST takes O(log n) time, where n is the number of nodes in the tree.
• Worst Case: If the BST is unbalanced (essentially becoming a linked list), searching can take O(n) time, where n is the number of nodes in the tree.

Path Length:

The path length in a binary tree refers to the length of the path from the root node to a particular node. In the context of binary search trees, the average path length is a measure of how balanced the tree is. A more balanced tree has a shorter average path length, leading to more efficient search operations.

AVL Trees:

AVL trees are self-balancing binary search trees named after their inventors, Adelson-Velsky and Landis. In an AVL tree, the heights of the two child subtrees of any node differ by at most one. This property helps maintain a balanced tree, which ensures that search, insertion, and deletion operations are all performed in O(log n) time on average.

To maintain the balance property, AVL trees perform rotations (left and right rotations) whenever necessary during insertion and deletion operations. These rotations ensure that the tree remains balanced and maintains the AVL property.

In an AVL tree, the height of the tree is kept relatively small compared to a non-balanced binary search tree, resulting in more efficient search operations. The worst-case time complexity for search, insertion, and deletion operations in an AVL tree is O(log n), making them suitable for applications where efficient search is crucial.

Overall, AVL trees are designed to address the issue of unbalanced binary search trees, providing efficient search operations by maintaining balance through rotations.