© 2015 Goodrich and Tamassia AVL Trees 1

AVL Trees 6

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Presentation for use with the textbook Algorithm Design and Applications, by M. T. Goodrich and R. Tamassia, Wiley, 2015

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AVL Tree Definition AVL trees are rank-balanced trees. The rank, r(v), of each node, v, is its height. Rank-balance rule: An AVL Tree is a binary search tree such that for every internal node v of T, the heights (ranks) of the children of v can differ by at most 1.

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An example of an AVL tree where the ranks are shown next to the nodes

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Height of an AVL Tree Fact: The height of an AVL tree storing n keys is O(log n). Proof (by induction): Let us bound n(h): the minimum number of internal nodes of an AVL tree of height h.

We easily see that n(1) = 1 and n(2) = 2 For n > 2, an AVL tree of height h contains the root node, one AVL subtree of height n-1 and another of height n-2. That is, n(h) = 1 + n(h-1) + n(h-2) Knowing n(h-1) > n(h-2), we get n(h) > 2n(h-2). So n(h) > 2n(h-2), n(h) > 4n(h-4), n(h) > 8n(n-6), … (by induction), n(h) > 2in(h-2i)

Solving the base case we get: n(h) > 2 h/2-1

Taking logarithms: h < 2log n(h) +2 Thus the height of an AVL tree is O(log n)

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Insertion Insertion is as in a binary search tree Always done by expanding an external node. Example:

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b=x

a=y

c=z

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before insertion

after insertion

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Trinode Restructuring Let (a,b,c) be the inorder listing of x, y, z Perform the rotations needed to make b the topmost node of the three

b=y

a=z

c=x T0

T1

T2 T3

b=y

a=z c=x

T0 T1 T2 T3

c=y

b=x

a=z

T0

T1 T2

T3 b=x

c=y a=z

T0 T1 T2 T3

Double rotation around c and a

Single rotation around b

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Insertion Example, continued

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T 0 T 1

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unbalanced...

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Restructuring (as Single Rotations)   Single Rotations:

T0T1

T2T3

c = xb = y

a = z

T0 T1 T2T3

c = xb = y

a = zsingle rotation

T3T2

T1T0

a = xb = y

c = z

T0T1T2T3

a = xb = y

c = zsingle rotation

© 2015 Goodrich and Tamassia AVL Trees 8

Restructuring (as Double Rotations) double rotations:

double rotationa = z

b = xc = y

T0T2

T1T3 T0

T2T3T1

a = zb = x

c = y

double rotationc = z

b = xa = y

T0T2

T1T3 T0

T2T3 T1

c = zb = x

a = y

© 2015 Goodrich and Tamassia

Pseudo-code Insertion.

AVL Trees 9

© 2015 Goodrich and Tamassia

Pseudo-code Rebalance at a node violating the rank rule.

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Removal Removal begins as in a binary search tree, which means the node removed will become an empty external node. Its parent, w, may cause an imbalance. Example:

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before deletion of 32 after deletion

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Rebalancing after a Removal Let z be the first unbalanced node encountered while travelling up the tree from w. Also, let y be the child of z with the larger height, and let x be the child of y with the larger height We perform a trinode restructuring to restore balance at z As this restructuring may upset the balance of another node higher in the tree, we must continue checking for balance until the root of T is reached

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w

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© 2015 Goodrich and Tamassia

Pseudo-code Removal

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AVL Tree Performance AVL tree storing n items n  The data structure uses O(n) space

n  A single restructuring takes O(1) time w  using a linked-structure binary tree

n  Searching takes O(log n) time w  height of tree is O(log n), no restructures needed

n  Insertion takes O(log n) time w  initial find is O(log n) w  restructuring up the tree, maintaining heights is O(log n)

n  Removal takes O(log n) time w  initial find is O(log n) w  restructuring up the tree, maintaining heights is O(log n)