(Balanced) Multi-way Search Tree

A multi-way search tree is an ordered tree. Each internal node has at least 2 child nodes and at most d (the degree of the tree) child nodes. If we visit the data items stored in the multi-way search tree in an augmented in-order traversal, then the keys of these data items are in some specific order.

An (a,b)-tree is a multiway search tree that satisfies the following requirements:

• Depth Property: All the leaf nodes are in the same level (at the same depth).
• Size Property I: Each internal node (except the root node) must have at least a child nodes and at most b child nodes.
• Size Property II: Each leaf node (except the root node) must have at least a-1 data items and at most b-1 data items.
• Size Property III: The number of data items stored in the internal nodes must be exactly one less than the number of its child nodes.
• Search Key Property: A slightly modified in-order traversal of the (a,b)-tree would visit the data items in the ascending order of their keys.

(2,4)-tree is an (a,b)-tree, where a = 2 and b = 4.

(2,4)-tree operations:

• lookup
• insert: always insert into a leaf node; handle the possible overflow by splitting; overflow may propagate to the parent node.
• remove: swap the victim to a leaf node then remove it; handle the possible underflow by borrowing or merging; underflow handled by merging may propagate to the parent node.

B-tree is an (a,b)-tree where a is about half of b.