Chapter 12
Multi-way Trees

12.1 Definition

A m-way search tree is a tree in which

1. The nodes hold between 1 to m-1 distinct keys
2. The keys in each node are sorted
3. A node with k values has k+1 subtrees, where the subtrees may be empty.
4. The i’th subtree of a node [v₁, ..., v_k], 0 < i < k, may hold only values v in the range v_i < v < v_i+1 (v₀ is assumed to equal -, and v_k+1 is assumed to equal infty).

A m-way tree of height h has between h and m^h - 1 keys.

12.2 Insertions

• Search the key going down the tree until reaching an empty subtree
• Insert the key to the parent of the empty subtree, if there is room in the node.

  insert 8

• Insert the key to the subtree, if there is no room in its parent.

  insert 27

12.3 Deletions

• If the key to be deleted is between two empty subtrees, delete it

  delete 8

• If the key is adjacent to a nonempty subtree, replace it with the largest key from the left subtree or the smallest key from the right subtree

  remove 16