AVL Trees: A Powerful Tool for Data Structures in Swift

AVL trees are a type of self-balancing binary search tree, 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, ensuring the tree remains balanced and allowing for efficient insertion, deletion, and search operations.

In Swift, we can implement an AVL tree by creating a Node class that has properties for the value, left child, right child, and height of the node. We can then create a AVLTree class that has a root Node and methods for inserting and removing nodes, as well as a method for checking if the tree is balanced.

To insert a new node into the tree, we first find the correct position for the node as in a standard binary search tree. Once we have found the correct position, we need to check if the tree has become unbalanced at the insertion point. If the tree is unbalanced, we perform a rotation to restore balance to the tree. There are four possible rotations that can be performed in an AVL tree: right rotation, left rotation, right-left rotation, and left-right rotation.

To remove a node from the tree, we first find the node to be removed and then replace it with the inorder successor (the next largest value in the tree). Once the node has been removed, we again need to check if the tree has become unbalanced at the removal point and perform a rotation if necessary to restore balance.

It is important to maintain the balance of an AVL tree as this ensures that the tree remains efficient and that the time complexity for operations such as insertion and deletion is O(log n).

It's about time to look at some examples:

Now, let's conduct a search, considering the number 5 for instance:

Now, let's test the deletion method:

Finally, if we conduct our search again on the number 5, we should anticipate a returned value of nil: