Weight-balanced tree

For other uses, see Optimal binary search tree.

In computer science, weight-balanced binary trees (WBTs) are a type of self-balancing binary search trees that can be used to implement dynamic sets, dictionaries (maps) and sequences.[1] These trees were introduced by Nievergelt and Reingold in the 1970s as trees of bounded balance, or BB[α] trees.[2][3] Their more common name is due to Knuth.[4]

A well known example is a Huffman coding of a corpus.

Like other self-balancing trees, WBTs store bookkeeping information pertaining to balance in their nodes and perform rotations to restore balance when it is disturbed by insertion or deletion operations. Specifically, each node stores the size of the subtree rooted at the node, and the sizes of left and right subtrees are kept within some factor of each other. Unlike the balance information in AVL trees (using information about the height of subtrees) and red–black trees (which store a fictional "color" bit), the bookkeeping information in a WBT is an actually useful property for applications: the number of elements in a tree is equal to the size of its root, and the size information is exactly the information needed to implement the operations of an order statistic tree, viz., getting the n'th largest element in a set or determining an element's index in sorted order.[5]

Weight-balanced trees are popular in the functional programming community and are used to implement sets and maps in MIT Scheme, SLIB, SML-NJ, and implementations of Haskell.[6][4]

  1. ^ Tsakalidis, A. K. (1984). "Maintaining order in a generalized linked list". Acta Informatica. 21: 101–112. doi:10.1007/BF00289142. S2CID 26127563.
  2. ^ Nievergelt, J.; Reingold, E. M. (1973). "Binary Search Trees of Bounded Balance". SIAM Journal on Computing. 2: 33–43. doi:10.1137/0202005.
  3. ^ Public Domain This article incorporates public domain material from Paul E. Black. "BB(α) tree". Dictionary of Algorithms and Data Structures. NIST.
  4. ^ a b Hirai, Y.; Yamamoto, K. (2011). "Balancing weight-balanced trees" (PDF). Journal of Functional Programming. 21 (3): 287. doi:10.1017/S0956796811000104.
  5. ^ Roura, Salvador (2001). A new method for balancing binary search trees. ICALP. Lecture Notes in Computer Science. Vol. 2076. pp. 469–480. doi:10.1007/3-540-48224-5_39. ISBN 978-3-540-42287-7.
  6. ^ Adams, Stephen (1993). "Functional Pearls: Efficient sets—a balancing act". Journal of Functional Programming. 3 (4): 553–561. doi:10.1017/S0956796800000885.