B-Tree

-trees were introduced by Bayer (1972) and McCreight. They are a special -ary balanced tree used in databases because their 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). A -tree satisfies the following properties:

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

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

3. Each path from the root 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).