Lecture 314:- 2-3 Trees

A 2-3 tree is a type of self-balancing search tree that allows 1, 2, or 3 keys per node and maintains certain properties to ensure balance. It is an example of a B-tree where each internal node can have either two keys and three children or one key and two children. 2-3 trees are used to implement dictionaries and associative arrays efficiently.

Properties of 2-3 Trees:

1. Every node has either one key or two keys.
2. Nodes with one key have two children, and nodes with two keys have three children.
3. All leaves of the tree are at the same level.
4. All the internal nodes with one key have keys in ascending order, and all the internal nodes with two keys have keys in ascending order.

Insertion in a 2-3 Tree: When inserting an element into a 2-3 tree, it first follows the usual binary search tree insertion method to find the appropriate position. If a node already has two keys and a new key needs to be inserted, the node splits into two nodes, and the middle key moves up to the parent node. This process might propagate up the tree if the parent node also already has two keys. This way, the tree remains balanced, and the height is kept in check.

Deletion in a 2-3 Tree: Deletion in a 2-3 tree is more complex than insertion. It involves handling several cases and merging or redistributing keys and nodes to maintain the properties of the 2-3 tree.

Advantages of 2-3 Trees:

1. Better Balance: 2-3 trees are balanced by design, ensuring that the height of the tree remains optimal, leading to efficient search, insertion, and deletion operations.
2. Improved Performance: Due to the balanced nature, the operations on 2-3 trees have better time complexities compared to unbalanced binary search trees.
3. Memory Efficiency: 2-3 trees require fewer pointers compared to other balanced trees like AVL trees or Red-Black trees, making them memory-efficient.

Disadvantages of 2-3 Trees:

1. Complexity: The implementation of 2-3 trees is more complex compared to simpler binary search trees. Insertion and deletion involve multiple cases to handle.
2. Slower Lookup: In practice, 2-3 trees may have slower lookup performance compared to other balanced trees like AVL trees or Red-Black trees.

2-3 trees are an important data structure in computer science and are widely used in various applications that require efficient dictionary or associative array operations. However, in practice, other self-balancing trees like Red-Black trees or AVL trees are more commonly used due to their simplicity and similar performance characteristics.