# AVL tree

(data structure)

**Definition:**
A *balanced* *binary search tree* where the *height* of the two subtrees (*children*) of a node differs by at most one. Look-up, insertion, and deletion are *O(log n)*, where n is the number of *nodes* in the tree.

**Generalization** (I am a kind of ...)

*height-balanced tree*, *balanced binary tree*, *binary search tree*, *red-black tree* (when colored).

**Aggregate child** (... is a part of or used in me.)

*left rotation*, *right rotation*.

**See also**
*B-tree*, *threaded tree*, *Fibonacci tree*.

*Note:
The structure is named for the inventors, Adelson-Velskii and Landis. If necessary, the tree is rebalanced after insertions or deletions using rotations. *

*
* After Gary Grubb <ggrubb@sr.hp.com>.

* An AVL tree is at least as balanced as a red-black tree.*

Author: PEB

## Implementation

Ben Pfaff's explanations and code (C). Maksim Goleta's C# Collections uses it to implement sorted set (C#). Bro. David Carlson's tutorial and code (C++). Richard McGraw's Navl-latest.tar.bz2 (C#). Worst-case behavior of traversal, annotated for real time (WOOP/ADA).
## More information

explanation and example.

**Georgii M. Adelson-Velskii** and **Evgenii M. Landis**, *An algorithm for the organization of information*, Doklady Akademii Nauk SSSR, 146:263-266, 1962 (Russian). English translation by Myron J. Ricci in Soviet Math. Doklady, 3:1259-1263, 1962.

(Doklady is Russian for "Report". Sometimes transliterated in English as Doclady or Dokladi.)

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Entry modified 12 November 2019.

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Paul E. Black, "AVL tree", in
*Dictionary of Algorithms and Data Structures* [online], Paul E. Black, ed. 12 November 2019. (accessed TODAY)
Available from: https://www.nist.gov/dads/HTML/avltree.html