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# Avl Trees Essay

621 words - 2 pages

Red-Black Trees 4/10/2002 11:1
AVL Trees 1
AVL Trees 6
3 8
4
v
z
AVL Trees 2
AVL Tree Definition
Š AVL trees are balanced. Š An AVL Tree is a
binary search tree such that for every internal node v of T, the heights of the children of v can differ by at most 1.
88
44
17 78
32 50
48 62
2
4
1
1
2
3
1
1
An example of an AVL tree where the heights are shown next to the nodes:
AVL Trees 3
Height of an AVL Tree Š Fact: The height of an AVL tree storing n keys is O(log n). Š Proof: Let us bound n(h): the minimum number of internal
nodes of an AVL tree of height h. Š We easily see that n(1) = 1 and n(2) = 2 Š 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. Š That is, n(h) = 1 + n(h-1) + n(h-2) Š 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)
Š Solving the base case we get: n(h) > 2 h/2-1 Š Taking logarithms: h < 2log n(h) +2 Š Thus the height of an AVL tree is O(log n)
3
4 n(1)
n(2)
AVL Trees 4
Insertion in an AVL Tree Š Insertion is as in a binary search tree Š Always done by expanding an external node. Š Example: 44
17 78
32 50 88
48 62
54 w
b=x
a=y
c=z
44
17 78
32 50 88
48 62
before insertion after insertion
AVL Trees 5
Trinode Restructuring Š let (a,b,c) be an inorder listing of x, y, z Š 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=ya=z
T0 T1 T2 T3 case 1: single rotation (a left rotation about a)
case 2: double rotation (a right rotation about c, then a left rotation about a)
(other two cases are symmetrical)
AVL Trees 6
Insertion Example, continued
88
44
17 78...

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