AVL Trees, AA Trees
Balancing Binary Search Trees, Rotations
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Table of Contents
1. AVL Trees
1.
Properties of AVL
2.
Rotations in AVL (Double Left, Double Right)
3.
AVL Insertion Algorithm
2. AA Trees
1.
Properties of AA
2.
Rotations in AA
1.
Skew
2.
Split
2
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3
AVL Trees
Properties and Rotations
AVL Tree
 AVL tree is a self-balancing binary-search tree (visualization)
 Height of two subtrees can differ by at most 1
17
9
5
25
12
Average
Space O(n)
Search O(log n)
Insert O(log n)
Delete O(log n)
Worst case
O(n)
O(log n)
O(log n)
O(log n)
5
AVL Tree Rebalancing
 Height difference is measured by a balance factor (BF)
 BF(Tree) = Height(Left) – Height(Right)
0
17
 BF of any node is in the range [-1, 1]
0
 If BF becomes -2 or 2  rebalance
9
25
1
0
0
5
12
20
0
6
AVL Tree Rebalancing
 Rebalancing is done by retracing
 Start from inserted node's parent
0-1
and go up to root
17
 Perform rotations to restore balance
0
9
25
12
0
0
5
20
12
19
01
0
7
Right Rotation
 Set
x
to be child of
 Set Right Child of
y
y
to be Left Child of
x
y
x
C
y
A
B
In Order
Preserved
Right rotation (x)
A
x
B
C
Left Rotation
 Set
y
to be child of
 Set Left Child of
x
x
to be Right Child of
In Order
Preserved
y
y
x
A
Left rotation (y)
x
B
C
y
C
A
B
AVL Tree Insertion Algorithm
1. Insert like in ordinary BST
4
20
2. Retrace up to root
1.
Modify balance / height
2.
If balance factor ∉ [-1,1]
 rebalance
2
3
12
25
2
2
8
17
1
1 1
4
1
10
26
1
15
19
10
Insertion - #1
 Insert 11
4
20
2
3
25
12
2
2
17
8
1
1
4
1
10
1
15
26
1
19
11
Insertion - #2
 11 <
20
 go left
4
20
2
3
25
12
2
2
17
8
1
1
4
1
10
1
15
26
1
19
12
Insertion - #3
 11 <
12
 go left
4
20
2
3
25
12
2
2
17
8
1
1
4
1
10
1
15
26
1
19
13
Insertion - #4
 11 >
8
 go right
4
20
2
3
25
12
2
2
17
8
1
1
4
1
10
1
15
26
1
19
14
Insertion - #5
 11 >
10
 go right
4
20
2
3
25
12
2
2
17
8
1
1
4
1
10
1
15
26
1
19
15
Insertion - #5
 Right node is null  insert
 Update

10
10
4
20
height
2
3
balance is -1
25
12
2
2
17
8
1
2
4
1
10
1
1
15
26
1
19
11
16
Insertion - #5
 Update

8
8
height
4
20
balance is -1
2
3
25
12
2
3
17
8
1
2
4
1
10
1
1
15
26
1
19
11
17
Insertion - #5
 Update

12
12
height
4
20
balance is 1
2
4
25
12
2
3
17
8
1
2
4
1
10
1
1
15
26
1
19
11
18
Insertion - #5
 Update
20
height

20
balance is 2

20
is left heavy
 rotate
20
5
20
2
4
25
12
right
2
3
17
8
1
2
4
1
10
1
1
15
26
1
19
11
19
Insertion - #5

17
Switch parent
4
12
5
3
20
8
1
4
2
2
17
10
1
2
1
11
15
25
1
19
1
26
20
Insertion - #5

20
Update height
4
12
3
3
20
8
1
4
2
2
17
10
1
2
1
11
15
25
1
19
1
26
21
Insertion - #5

12

12
Update height
4
12
Balance is 0
3
3
20
8
1
4
2
2
17
10
1
2
1
11
15
25
1
19
1
26
22
Insertion - #5
 Over
4
12
3
3
20
8
1
4
2
2
17
10
1
2
1
11
15
More: https://en.wikipedia.org/wiki/AVL_tree
25
1
19
1
26
23
AVL Tree - Quiz
TIME’S UP!
TIME:
 Delete 25. What will be the resulting tree?
17
9
5
25
12
24
AVL Tree - Quiz
TIME’S UP!
 Delete 25. What will be the resulting tree?
17
9
5
17
25
12


9
5
9
12
5
17
12
25
Double Rotations
Double Left, Double Right Rotation
Single Rotation Problem
 Insert 4
2
5
-1
D
3
0
A
4
B
C
27
Single Rotation Problem (2)
 Rotate a node with opposite balanced child
-2
2
5
-1
3
D
3
0
A
1
A
5
0
D
4
4
B
Rotate Right 5
doesn’t solve it
C
B
C
28
Double Right Rotation
Right-Left
Double Right Rotation
 Rotate Right (node) with negatively balanced Left Child
2
5
-1
D
3
0
A
4
B
C
30
Double Right Rotation
 Left Rotate 3
2
5
-1
D
3
0
A
4
B
C
31
AVL Tree - Double Rotations
 Update Balance 3
2
5
0
D
4
-1
C
3
A
B
32
AVL Tree - Double Rotations
 Update Balance 3
2
5
0
D
4
0
C
3
A
B
33
AVL Tree - Double Rotations
 Right Rotate (5)
2
5
1
D
4
0
C
3
A
Reduced to
Single Right (5)
B
34
AVL Tree - Double Rotations
 Update Balance 5
1
4
0
2
3
A
5
B
C
D
35
AVL Tree - Double Rotations
 Update Balance 4
1
4
0
0
3
A
5
B
C
D
36
AVL Tree - Double Rotations
 Balance Restored!
0
4
0
0
3
A
5
B
C
D
37
Double Left Rotation
Left-Right
AVL Tree - Double Rotations
 Rotate Left (node) with positively balanced Right Child
-2
5
1
D
0
3
A
4
C
B
39
AVL Tree - Double Rotations
 Rotate Right (3)
-2
5
1
D
0
3
A
4
C
B
40
AVL Tree - Double Rotations
 Update Balance (3)
-2
5
0
D
4
1
C
3
B
A
41
AVL Tree - Double Rotations
 Update Balance (3)
-2
5
0
D
4
0
C
3
B
A
42
AVL Tree - Double Rotations
 Rotate Left (5)
-2
Reduced to
Single Left (5)
5
1
D
4
0
C
3
B
A
43
AVL Tree - Double Rotations
 Update Balance (5)
1
4
-2
0
3
5
D
C
C
A
44
AVL Tree - Double Rotations
 Update Balance (5)
1
4
0
0
3
5
D
C
C
A
45
AVL Tree - Double Rotations
 Update Balance (4)
0
4
0
0
3
5
D
C
C
A
46
AVL Tree - Quiz
TIME’S UP!
TIME:
 Insert 22. What will be the resulting tree?
17
9
5
25
12
20
47
AVL Tree - Quiz
TIME’S UP!
 Insert 22. What will be the resulting tree?
17
17
9
5
25
12
20

17
9
5
25
12

9
5
20
22
12
20
25
22
48
AVL Tree - Summary
Structure
Worst case
Average case
Search
Insert
Delete
BST
N
N
N
2-3 Tree
c lg N
c lg N
c lg N
c lg N
c lg N
Red-Black
2 lg N
2 lg N
2 lg N
lg N
lg N
lg N
lg N
AVL Tree
1.44 lg N 1.44 lg N 1.44 lg N
Search Hit
Insert
1.39 lg N 1.39 lg N
Insert/Delete perform
O(lgN) rotations
49
17
9
5
25
12
20
22
AVL Tree
Balancing Implementation
50
Rotate Right
public static Node<T> RotateRight(Node<T> node)
{
Node<T> left = node.Left;
node.Left = node.Left.Right;
left.Right = node;
UpdateHeight(node);
return left;
}
51
Balance Node
private static Node<T> Balance(Node<T> node)
{
int balance = Height(node.Left) - Height(node.Right);
if (balance < -1) // Right child is heavy
{
balance = Height(node.Right.Left) - Height(node.Right.Right);
if (balance <= 0) { return RotateLeft(node); }
else { node.Right = RotateRight(node.Right); return RotateLeft(node); }
}
else if (balance > 1) // Left child is heavy
{
balance = Height(node.Left.Left) - Height(node.Left.Right);
if (balance >= 0) { return RotateRight(node); }
else { node.Left = RotateLeft(node.Left); return RotateRight(node); }
}
return node;
}
52
Lab Exercise
AVL Tree - Balancing
AA Tree
 Utilizes the concept of levels
 Level - the number of left links on the path to a null node
Level 3
17
9
5
Level 2
25
12
22
32
Level 1
54
AA Tree
 AA tree invariants
 The level of every leaf node is one
 Every left child has level one less than its parent
 Every right child has level equal to or one less than its parent
 Right grandchildren have levels less than their grandparents
 Every node of level greater than one has two children
55
AA Tree
 Right horizontal links are possible
 Left horizontal links are not allowed
Level 3
17
9
5
Level 2
25
10
12
22
32
55
Level 1
56
Skew
 Skew operation is a single right rotation
 Skew when an insertion or deletion creates a horizontal left link
Level 3
17
9
5
Level 2
25
10
12
22
32
55
Level 1
57
Skew (2)
 Skew operation is a single right rotation
 Skew when an insertion or deletion creates a horizontal left link
Level 3
17
9
5
Level 2
25
10
12
22
32
55
Level 1
58
Split
 Split operation is a single left rotation
 Split when an insertion or deletion two consecutive right
horizontal links
Level 3
17
9
5
Level 2
25
10
12
22
32
55
71
Level 1
59
Split
 Split operation is a single left rotation
 Split when an insertion or deletion two consecutive right
horizontal links
Level 3
17
9
5
10
12
22
Level 2
55
25
32
71
Level 1
60
17
9
5
55
25
10
12
22
32
71
AA Tree
Insertion Algorithm
61
AA Tree Insertion #1
 Insert: 6
 New nodes are always inserted at Level 1
Level 3
Level 2
6
Level 1
62
AA Tree Insertion #1
 Insert: 2
Level 3
Level 2
6
Level 1
63
AA Tree Insertion #1
 Left horizontal link is not allowed
 Rotate
6
right (skew)
Level 3
Level 2
2
6
Level 1
64
AA Tree Insertion #1
 Insert: 8
Level 3
Level 2
2
6
Level 1
65
AA Tree Insertion #1
 Insert: 8
Level 3
Level 2
2
6
Level 1
66
AA Tree Insertion #1
 Insert: 8
Level 3
Level 2
2
6
Level 1
67
AA Tree Insertion #1
 Two consecutive right horizontal links
 Rotate
2
left (split)
Level 3
Level 2
2
6
8
Level 1
68
AA Tree Insertion #1
 Insert: 16
Level 3
Level 2
6
2
8
Level 1
69
AA Tree Insertion #1
 Insert: 16
Level 3
Level 2
6
2
8
Level 1
70
AA Tree Insertion #1
 Insert: 16
Level 3
Level 2
6
2
8
Level 1
71
AA Tree Insertion #1
 Insert: 16
Level 3
Level 2
6
2
8
16
Level 1
72
AA Tree Insertion #1
 Insert: 10
Level 3
Level 2
6
2
8
16
Level 1
73
AA Tree Insertion #1
 Insert: 10
Level 3
Level 2
6
2
8
16
Level 1
74
AA Tree Insertion #1
 Insert: 10
Level 3
Level 2
6
2
8
16
Level 1
75
AA Tree Insertion #1
 Horizontal left link not allowed
 Rotate
16
right (skew)
Level 3
Level 2
6
2
8
16
Level 1
10
76
AA Tree Insertion #1
 Two consecutive right horizontal links
 Rotate
8
left (split)
Level 3
Level 2
6
2
8
10
16
Level 1
77
AA Tree Insertion #1
 Two consecutive right horizontal links
 Rotate
8
left (split)
Level 3
6
2
Level 2
10
8
16
Level 1
78
Trees - Quiz
TIME’S UP!
TIME:
 Consider web application in which
 searches are far more frequent than insertions/deletions
 Which of the following do you prefer:
 AVL
 Linked List
 Red-Black
 B+
79
Trees - Quiz
TIME’S UP!
 Consider web application in which
 searches are far more frequent than insertions/deletions
 Which of the following do you prefer:
 AVL
 Linked List
AVL trees are more
rigidly balanced, so they
have faster search
 Red-Black
 B+
80
Summary
 AVL/AA trees tend to be flatter than Red-Black Trees
 AVL/AA trees perform more rotations for insertion and deletion
 Red-Black trees have faster insertion/deletion
 AVL/AA trees have faster searching
 AA have simpler implementation
81
AVL Trees, AA Trees
?
https://softuni.bg/trainings/1308/data-structures-february-2016
License
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is licensed under the "Creative Commons AttributionNonCommercial-ShareAlike 4.0 International" license
 Attribution: this work may contain portions from

"Fundamentals of Computer Programming with C#" book by Svetlin Nakov & Co. under CC-BY-SA license

"Data Structures and Algorithms" course by Telerik Academy under CC-BY-NC-SA license
83
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