Lecture 315:- 2-3-4 Trees

A 2-3-4 tree, also known as a 2-4 tree, is a type of self-balancing search tree and an extension of the 2-3 tree. It allows 2, 3, or 4 keys per node and maintains certain properties to ensure balance. 2-3-4 trees are used to implement dictionaries and associative arrays efficiently, similar to 2-3 trees.

Properties of 2-3-4 Trees:

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

Insertion in a 2-3-4 Tree: Insertion in a 2-3-4 tree follows a similar approach to a 2-3 tree. When inserting an element, it first follows the usual binary search tree insertion method to find the appropriate position. If a node already has three 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 three keys. This way, the tree remains balanced and maintains the 2-3-4 tree properties.

Deletion in a 2-3-4 Tree: Deletion in a 2-3-4 tree is more complex than insertion. It involves handling several cases and merging or redistributing keys and nodes to maintain the properties of the 2-3-4 tree, similar to deletion in a 2-3 tree.

Advantages of 2-3-4 Trees:

1. Better Balance: 2-3-4 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 2-3-4 trees have better time complexities compared to unbalanced binary search trees.
3. Memory Efficiency: 2-3-4 trees require fewer pointers compared to other balanced trees like AVL trees or Red-Black trees, making them memory-efficient.

Disadvantages of 2-3-4 Trees:

1. Complexity: The implementation of 2-3-4 trees is more complex compared to simpler binary search trees. Insertion and deletion involve multiple cases to handle.
2. Slower Lookup: In practice, 2-3-4 trees may have slower lookup performance compared to other balanced trees like AVL trees or Red-Black trees.

2-3-4 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, like 2-3 trees, 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.