COMP171

Fall 2005

AVL-Trees
AVL Trees / Slide 2

Balanced binary tree

 The disadvantage of a binary search tree is that its
height can be as large as N-1
 This means that the time needed to perform insertion
and deletion and many other operations can be O(N)
in the worst case
 We want a tree with small height
 A binary tree with N node has height at least (log N)
 Thus, our goal is to keep the height of a binary search
tree O(log N)
 Such trees are called balanced binary search trees.
Examples are AVL tree, red-black tree.
AVL Trees / Slide 3

AVL tree
Height of a node
 The height of a leaf is 1. The height of a null
pointer is zero.
 The height of an internal node is the
maximum height of its children plus 1

Note that this definition of height is different from the

one we defined previously (we defined the height of a
leaf as zero previously).
AVL Trees / Slide 4

AVL tree
 An AVL tree is a binary search tree in which
 for every node in the tree, the height of the left and
right subtrees differ by at most 1.

AVL property
violated here
AVL Trees / Slide 5

AVL tree
 Let x be the root of an AVL tree of height h
 Let Nh denote the minimum number of nodes in an
AVL tree of height h
 Clearly, Ni ≥ Ni-1 by definition
 We have N h  N h 1  N h  2  1
 2 N h2  1
 2 N h2
 By repeated substitution, we obtain the general form
N h  2i N h  2
 The boundary conditions are: N1=1 and N2 =2. This
implies that h = O(log Nh).
 Thus, many operations (searching, insertion, deletion)
on an AVL tree will take O(log N) time.
AVL Trees / Slide 6

Rotations
 When the tree structure changes (e.g., insertion or
deletion), we need to transform the tree to restore the
AVL tree property.
 This is done using single rotations or double rotations.

e.g. Single Rotation

y
x
x
y C
A C
B
B
A After Rotation
Before Rotation
AVL Trees / Slide 7

Rotations
 Since an insertion/deletion involves
adding/deleting a single node, this can only
increase/decrease the height of some subtree
by 1
 Thus, if the AVL tree property is violated at a
node x, it means that the heights of left(x) ad
right(x) differ by exactly 2.
 Rotations will be applied to x to restore the
AVL tree property.
AVL Trees / Slide 8

Insertion
 First, insert the new key as a new leaf just as in
ordinary binary search tree
 Then trace the path from the new leaf towards the
root. For each node x encountered, check if heights
of left(x) and right(x) differ by at most 1.
 If yes, proceed to parent(x). If not, restructure by
doing either a single rotation or a double rotation [next
slide].
 For insertion, once we perform a rotation at a node x,
we won’t need to perform any rotation at any ancestor
of x.
AVL Trees / Slide 9

Insertion
 Let x be the node at which left(x) and right(x)
differ by more than 1
 Assume that the height of x is h+3
 There are 4 cases
 Height of left(x) is h+2 (i.e. height of right(x) is h)
Height of left(left(x)) is h+1  single rotate with left child
Height of right(left(x)) is h+1  double rotate with left child
 Height of right(x) is h+2 (i.e. height of left(x) is h)
Height of right(right(x)) is h+1  single rotate with right child
Height of left(right(x)) is h+1  double rotate with right child

Note: Our test conditions for the 4 cases are different from the code shown in the
textbook. These conditions allow a uniform treatment between insertion and deletion.
AVL Trees / Slide 10

Single rotation
The new key is inserted in the subtree A.
The AVL-property is violated at x
 height of left(x) is h+2
 height of right(x) is h.
AVL Trees / Slide 11

Single rotation
The new key is inserted in the subtree C.
The AVL-property is violated at x.

Single rotation takes O(1) time.

Insertion takes O(log N) time.
AVL Trees / Slide 12

AVL Tree 5 x
5

3 y 8 C
3 8

1 4
1 4 B
A
0.8

Insert 0.8
3

1
5

8
After rotation
4
0.8
AVL Trees / Slide 13

Double rotation
The new key is inserted in the subtree B1 or B2.
The AVL-property is violated at x.
x-y-z forms a zig-zag shape

also called left-right rotate

AVL Trees / Slide 14

Double rotation
The new key is inserted in the subtree B1 or B2.
The AVL-property is violated at x.

also called right-left rotate

AVL Trees / Slide 15

5 x
AVL Tree 5

y C
8
3
3 8

A 1 4 z
1 4

B 3.5

Insert 3.5
4

3
5

3.5 8 After Rotation

1
AVL Trees / Slide 16

An Extended Example

Insert 3,2,1,4,5,6,7, 16,15,14 Single rotation

3 2
3
3

2 1
Fig 1 2 3
Fig 4
Fig 2
2 1 2
Single rotation
Fig 3
1
1 3
3
Fig 5 Fig 6 4
4
5
AVL Trees / Slide 17
2
2
Single rotation
1
1
4 4
3 5
3 5
Fig 8
Fig 7 6
4 4

Single rotation
2 2
5 5
1 3 6 1 3 6
4

Fig 9 Fig 10 7
2
6
1 3 7

5 Fig 11
AVL Trees / Slide 18

2
6
1 3 7

5 16
Fig 12

4
Double rotation 4
2
6
2
7 6
1 3
1 3 15
5 16 5
16
Fig 13 7
15 Fig 14
AVL Trees / Slide 19

4 4
Double rotation

2 2 7
6
15 1 3 15
1 3 5 6
16
7 5 14 16
Fig 15
14 Fig 16
AVL Trees / Slide 20

Deletion
 Delete a node x as in ordinary binary search tree.
Note that the last node deleted is a leaf.
 Then trace the path from the new leaf towards the
root.
 For each node x encountered, check if heights of
left(x) and right(x) differ by at most 1. If yes, proceed
to parent(x). If not, perform an appropriate rotation at
x. There are 4 cases as in the case of insertion.
 For deletion, after we perform a rotation at x, we may
have to perform a rotation at some ancestor of x.
Thus, we must continue to trace the path until we
reach the root.
AVL Trees / Slide 21

Deletion
 On closer examination: the single rotations for
deletion can be divided into 4 cases (instead
of 2 cases)
 Two cases for rotate with left child
 Two cases for rotate with right child
AVL Trees / Slide 22

Single rotations in deletion

In both figures, a node is deleted in subtree C, causing the height
to drop to h. The height of y is h+2. When the height of subtree A
is h+1, the height of B can be h or h+1. Fortunately, the same
single rotation can correct both cases.

rotate with left child

AVL Trees / Slide 23

Single rotations in deletion

In both figures, a node is deleted in subtree A, causing the height
to drop to h. The height of y is h+2. When the height of subtree C
is h+1, the height of B can be h or h+1. A single rotation can
correct both cases.

rotate with right child

AVL Trees / Slide 24

Rotations in deletion
 There are 4 cases for single rotations, but we
do not need to distinguish among them.
 There are exactly two cases for double
rotations (as in the case of insertion)
 Therefore, we can reuse exactly the same
procedure for insertion to determine which
rotation to perform