Chapter notes

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## Chapter notes

The idea of balancing a search tree is due to Adel'son-Vel'ski i and Landis [2], who introduced a class of balanced search trees called "AVL trees" in 1962, described in Problem 13-3. Another class of search trees, called "2-3 trees," was introduced by J. E. Hopcroft (unpublished) in 1970. Balance is maintained in a 2-3 tree by manipulating the degrees of nodes in the tree. A generalization of 2-3 trees introduced by Bayer and McCreight [32], called B-trees, is the topic of Chapter 18.

Red-black trees were invented by Bayer [31] under the name "symmetric binary B-trees." Guibas and Sedgewick [135] studied their properties at length and introduced the red/black color convention. Andersson [15] gives a simpler-to-code variant of red-black trees. Weiss [311] calls this variant AA-trees. An AA-tree is similar to a red-black tree except that left children may never be red.

Treaps were proposed by Seidel and Aragon [271]. They are the default implementation of a dictionary in LEDA, which is a well-implemented collection of data structures and algorithms.

There are many other variations on balanced binary trees, including weight-balanced trees [230], k-neighbor trees [213], and scapegoat trees [108]. Perhaps the most intriguing are the "splay trees" introduced by Sleator and Tarjan [281], which are "self-adjusting." (A good description of splay trees is given by Tarjan [292].) Splay trees maintain balance without any explicit balance condition such as color. Instead, "splay operations" (which involve rotations) are performed within the tree every time an access is made. The amortized cost (see Chapter 17) of each operation on an n-node tree is O(lg n).

Skip lists [251] are an alternative to balanced binary trees. A skip list is a linked list that is augmented with a number of additional pointers. Each dictionary operation runs in expected time O(lg n) on a skip list of n items.

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