B-Tree

A -tree is a special balanced -ary tree used in databases because its structure allows records to be inserted, deleted, and retrieved with guaranteed worst-case performance. An -node -tree has height , where lg is the logarithm to base 2. The Apple® Macintosh® (Apple, Inc., Cupertino, CA) HFS filing system uses -trees to store disk directories (Benedict 1995). -trees were introduced by Bayer and McCreight (1972). A -tree satisfies the following properties:

1. The root vertex is either a tree leaf or has at least two children.

2. Each node (except the root vertex and tree leaves) has between and children, where is the ceiling function.

3. Each graph path from the root vertex to a tree leaf has the same length.

Every 2-3 tree is a -tree of order 3. The number of -trees of order 3 with , 2, ... leaves are 1, 1, 1, 1, 2, 2, 3, 4, 5, 8, 14, 23, 32, 43, 63, ... (Ruskey, OEIS A014535). The number of order-4 -trees with , 2, ... leaves are 1, 1, 1, 2, 2, 4, 5, 9, 15, 28, 45, ... (OEIS A037026).