AVL tree

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AVL tree
Type	tree
Invented	1962
Invented by	Georgy Adelson-Velsky and Evgenii Landis
Time complexity in big O notation Operation Average Worst case Search O(log n)[1] O(log n)[1] Insert O(log n)[1] O(log n)[1] Delete O(log n)[1] O(log n)[1] Space complexity Space O(n) O(n)

In

.

The AVL tree is named after its two Soviet inventors, Georgy Adelson-Velsky and Evgenii Landis, who published it in their 1962 paper "An algorithm for the organization of information".^[3]

AVL trees are often compared with red–black trees because both support the same set of operations and take O(log n) time for the basic operations. For lookup-intensive applications, AVL trees are faster than red–black trees because they are more rigidly balanced.^[4] Similar to red–black trees, AVL trees are height-balanced. Both are in general not weight-balanced nor μ-balanced for any μ≤¹⁄₂;^[5] that is, sibling nodes can have hugely differing numbers of descendants.

Languages

1. AVL Trees in Java
2. AVL Trees in C#
3. AVL Trees in C++

Definition

Balance factor

In a binary tree the balance factor of a node N is defined to be the height difference

BalanceFactor(N) := –Height(LeftSubtree(N)) + Height(RightSubtree(N)) ^[6]

of its two child subtrees. A binary tree is called AVL tree if the

BalanceFactor(N) ∈ {–1,0,+1}

holds for every node N in the tree.

A node N with BalanceFactor(N) < 0 is called "left-heavy", one with BalanceFactor(N) > 0 is called "right-heavy", and one with BalanceFactor(N) = 0 is sometimes simply called "balanced".

Remark

In the sequel, because there is a one-to-one correspondence between nodes and the subtrees rooted by them, we sometimes let it to the context whether the name of an object stands for the node or the subtree.

Properties

Balance factors can be kept up-to-date by knowing the previous balance factors and the change in height – it is not necessary to know the absolute height. For holding the AVL balance information, two bits per node are sufficient.^[7]

The height h of an AVL tree with n nodes lies in the interval:^[8]

log₂(n+1) ≤ h < c log₂(n+2)+b

with the golden ratio φ := (1+√5) ⁄₂ ≈ 1.618, c := ¹⁄ log₂ φ ≈ 1.44,  and  b := ^c⁄₂ log₂ 5 – 2 ≈ –0.328. This is because an AVL tree of height h contains at least F_h+2 – 1 nodes where {F_h} is the

with the seed values F₁ = 1, F₂ = 1.

Data structure

According to the original paper "An algorithm for the organization of information" AVL trees have been invented as binary search trees. In that sense they are a data structure together with its major associated operations, namely search, insert, delete, which rely on and maintain the AVL property. In that sense, the AVL tree is a "self-balancing binary search tree".

Operations

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Read-only operations of an AVL tree involve carrying out the same actions as would be carried out on an unbalanced binary search tree, but modifications have to observe and restore the height balance of the subtrees.

Searching

Searching for a specific key in an AVL tree can be done the same way as that of a normal unbalanced binary search tree. In order for search to work effectively it has to employ a comparison function which establishes a total order (or at least a total preorder) on the set of keys. The number of comparisons required for successful search is limited by the height h and for unsuccessful search is very close to h, so both are in O(log n).

Traversal

Once a node has been found in an AVL tree, the next or previous node can be accessed in

constant time. Some instances of exploring these "nearby" nodes require traversing up to h ∝ log(n) links (particularly when navigating from the rightmost leaf of the root’s left subtree to the root or from the root to the leftmost leaf of the root’s right subtree; in the AVL tree of figure 1, moving from node P to the next but one node Q takes 3 steps). However, exploring all n nodes of the tree in this manner would visit each link exactly twice: one downward visit to enter the subtree rooted by that node, another visit upward to leave that node’s subtree after having explored it. And since there are n−1 links in any tree, the amortized cost is found to be 2×(n−1)/n, or approximately 2.

Comparison to other structures

Both AVL trees and red–black trees are self-balancing binary search trees and they are very similar mathematically.^[9] The operations to balance the trees are different, but both occur on the average in O(1) with maximum in O(log n). The real difference between the two is the limiting height.

For a tree of size n ≥ 1

• an AVL tree’s height is at most

  ${\displaystyle {\begin{array}{ll}h&\leqq \;c\log _{2}(n+d)+b\\&<\;c\log _{2}(n+2)+b\end{array}}}$

where ${\displaystyle \varphi :={\tfrac {1+{\sqrt {5}}}{2}}\approx 1.618}$  the golden ratio, ${\displaystyle c:={\tfrac {1}{\log _{2}\varphi }}\approx 1.44,}$   ${\displaystyle b:={\tfrac {c}{2}}\log _{2}5-2\approx \;-0.328,}$ and  ${\displaystyle d:=1+{\tfrac {1}{\varphi ^{4}{\sqrt {5}}}}\approx 1.07}$.

• a red–black tree’s height is at most

  ${\displaystyle {\begin{array}{ll}h&\leqq \;2\log _{2}(n+1)\end{array}}}$ ^[10]

AVL trees are more rigidly balanced than red–black trees, leading to faster retrieval but slower insertion and deletion.

See also

• Trees
• Tree rotation
• Red–black tree
• Splay tree
• Scapegoat tree
• B-tree
• T-tree
• List of data structures

References

Further reading

• Donald Knuth. The Art of Computer Programming, Volume 3: Sorting and Searching, Third Edition. Addison-Wesley, 1997. ISBN 0-201-89685-0. Pages 458–475 of section 6.2.3: Balanced Trees.

External links

The Wikibook Algorithm Implementation has a page on the topic of: AVL tree

Wikimedia Commons has media related to AVL-trees.

•  This article incorporates
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• AVL tree demonstration (requires Flash)
• AVL tree demonstration (requires Java)