What is a 2-3 tree, and how does it work?

A 2-3 tree is a type of balanced search tree where every node can have either two children (2-node) or three children (3-node).

A 2-3 tree is a specific type of B-tree that is self-balancing, meaning it automatically adjusts its structure to maintain balance as elements are added or removed. This makes it an efficient data structure for operations like searching, insertion, and deletion. The '2-3' in its name refers to the number of children nodes each parent node can have. A 2-node has one key and two children, where the left child is less than the key and the right child is greater. A 3-node has two keys and three children, where the left child is less than both keys, the middle child is between the keys, and the right child is greater than both keys.

The process of inserting a new element into a 2-3 tree involves finding the appropriate leaf node where the element should be placed, based on its value. If the leaf node is a 2-node, it can simply be expanded to a 3-node by adding the new element. However, if the leaf node is already a 3-node, it must be split into two 2-nodes, and the middle key is moved up to the parent node. This process may need to be repeated if the parent node is also a 3-node, potentially all the way up to the root of the tree.

Deletion in a 2-3 tree is slightly more complex. If the key to be deleted is in a leaf node, it can simply be removed. If the key is in an internal node, it is replaced with its predecessor or successor (which is guaranteed to be in a leaf node), and then that leaf node is deleted. If deleting a key would leave a node with no keys, the tree is restructured to ensure that every node remains a 2-node or 3-node.

In summary, a 2-3 tree is a versatile and efficient data structure that maintains balance to ensure that operations like searching, insertion, and deletion can be performed quickly. Its structure, where each node can have two or three children, allows it to adapt dynamically as elements are added or removed.