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).

REFERENCES

Aho, A. V.; Hopcroft, J. E.; and Ullmann, J. D. Data Structures and Algorithms. Reading, MA: Addison-Wesley, pp. 369-374, 1987.

Bayer, R. and McCreight, E. "Organization and Maintenance of Large Ordered Indexes." Acta Informatica 1, 173-189, 1972.

Benedict, B. Using Norton Utilities for the Macintosh. Indianapolis, IN: Que, pp. B-17-B-33, 1995.

Beyer, R. "Symmetric Binary -Trees: Data Structures and Maintenance Algorithms." Acta Informat. 1, 290-306, 1972.

Knuth, D. E. "B-Trees." The Art of Computer Programming, Vol. 3: Sorting and Searching, 2nd ed. Reading, MA: Addison-Wesley, pp. 482-485 and 490-491, 1998.

Ruskey, F. "Information on B-Trees." http://www.theory.csc.uvic.ca/~cos/inf/tree/BTrees.html.Skiena, S. S. The Algorithm Design Manual. New York: Springer-Verlag, p. 178, 1997.

Sloane, N. J. A. Sequences A014535 and A037026 in "The On-Line Encyclopedia of Integer Sequences."