1. Balancing Binary Search Trees, Rotations
AVL Trees, AA Trees
Balancing Binary Search Trees, Rotations
Tree-Like Structures
SoftUni Team
Technical Trainers
Software University
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2. Table of Contents AVL Trees AA Trees Properties of AVL*
Table of Contents
AVL Trees
Properties of AVL
Rotations in AVL (Double Left, Double Right)
AVL Insertion Algorithm
AA Trees
Properties of AA
Rotations in AA
Skew
Split
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3. Have a Question?
sli.do #DsAlgo

4. Properties and Rotations
AVL Trees
Properties and Rotations

5. AVL Tree
AVL tree is a self-balancing binary-search tree (visualization)
Height of two subtrees can differ by at most 1 17 25 9 5 12
Average
Worst case
Space
O(n)
Search
O(log n)
Insert
Delete

6. AVL Tree Rebalancing
Height difference is measured by a balance factor (BF)
BF(Tree) = Height(Left) – Height(Right)
BF of any node is in the range [-1, 1]
If BF becomes -2 or 2  rebalance 17 25 9 5 12 20 1

7. AVL Tree Rebalancing Rebalancing is done by retracing
Start from inserted node's parent and go up to root
Perform rotations to restore balance -1 17 25 9 5 12 20 19 1 2 1

8. Right Rotation Set to be child ofx y
Set to be child of
Set Right Child of to be Left Child of y x
In Order Preserved x y C A
Right rotation (x) x y A B B C

9. Left Rotation Set to be child ofy x
Set to be child of
Set Left Child of to be Right Child of x y
In Order Preserved y x A C
Left rotation (y) x y B C A B

10. AVL Tree Insertion Algorithm
Insert like in ordinary BST
Retrace up to root
Modify balance / height
If balance factor ∉ [-1,1]  rebalance 4 20 3 2 12 25 2 2 1 8 17 26 1 1 1 1 4 10 15 19

11. Insertion - #1
Insert 11 4 20 3 2 12 25 2 2 1 8 17 26 1 1 1 1 4 10 15 19

12. Insertion - #2 11 <  go left 4 2 3 2 1 2 1 1 1 1 20 20 12 25 8 1726 1 1 1 1 4 10 15 19

13. Insertion - #3 11 <  go left 4 2 3 2 1 2 1 1 1 1 12 20 12 25 8 1726 1 1 1 1 4 10 15 19

14. Insertion - #4 11 >  go right 4 2 3 2 1 2 1 1 1 1 8 20 12 25 8 1726 1 1 1 1 4 10 15 19

15. Insertion - #5 11 >  go right 4 2 3 2 1 2 1 1 1 1 10 20 12 25 8 1726 1 1 1 1 4 10 15 19

16. Insertion - #5 Right node is null  insert Update height balance is -14 20 10 3 2 10 12 25 2 2 1 8 17 26 1 2 1 1 4 10 15 19 1 11

17. Insertion - #5 Update height balance is -1 4 2 3 2 1 3 1 1 2 1 1 8 2012 25 2 3 1 8 17 26 1 2 1 1 4 10 15 19 1 11

18. Insertion - #5 Update height balance is 1 4 2 4 2 1 3 1 1 2 1 1 12 2025 2 3 1 8 17 26 1 2 1 1 4 10 15 19 1 11

19. Insertion - #5 Update height balance is 2 is left heavy rotate right 520 5 20 20 4 2 20 12 25 2 20 3 1 8 17 26 1 2 1 1 4 10 15 19 1 11

20. Insertion - #5 Switch parent 4 5 3 2 2 1 2 1 1 1 1 17 12 20 8 17 25 410 1 1 1 1 11 15 19 26

21. Insertion - #5 Update height 4 3 3 2 2 1 2 1 1 1 1 20 12 20 8 17 25 410 1 1 1 1 11 15 19 26

22. Insertion - #5 Update height Balance is 0 4 3 3 2 2 1 2 1 1 1 1 12 1220 8 2 2 1 2 17 25 4 10 1 1 1 1 11 15 19 26

23. Insertion - #5 Over More: https://en.wikipedia.org/wiki/AVL_tree 4 3 312 3 3 20 8 2 2 1 2 17 25 4 10 1 1 1 1 11 15 19 26

24. AVL Tree - Quiz Delete 25. What will be the resulting tree? 17 25 9 5
TIME:
TIME’S UP!
Delete 25. What will be the resulting tree? 17 25 9 5 12

25. AVL Tree - Quiz   Delete 25. What will be the resulting tree? 17 25
TIME’S UP!
Delete 25. What will be the resulting tree? 17 25 9 5 12 17 9 5 12 9 5 17 12  

26. Double Left, Double Right Rotation
Double Rotations
Double Left, Double Right Rotation

27. Single Rotation Problem
Insert 4 2 5 D -1 3 A 4 B C

28. Single Rotation Problem (2)
Rotate a node with opposite balanced child 2 -2
Rotate Right 5 doesn’t solve it 5 3 D A 1 -1 3 5 A D 4 4 B C B C

29. Double Right Rotation
Right-Left

30. Double Right Rotation
Rotate Right (node) with negatively balanced Left Child 2 5 D -1 3 A 4 B C

31. Double Right Rotation
Left Rotate 3 2 5 D -1 3 A 4 B C

32. AVL Tree - Double Rotations
Update Balance 3 2 5 D 4 C -1 3 A B

33. AVL Tree - Double Rotations
Update Balance 3 2 5 D 4 C 3 A B

34. AVL Tree - Double Rotations
Right Rotate (5) 2
Reduced to Single Right (5) 5 D 1 4 C 3 A B

35. AVL Tree - Double Rotations
Update Balance 5 1 4 2 3 5 A B C D

36. AVL Tree - Double Rotations
Update Balance 4 1 4 3 5 A B C D

37. AVL Tree - Double Rotations
Balance Restored! 4 3 5 A B C D

38. Double Left Rotation
Left-Right

39. AVL Tree - Double Rotations
Rotate Left (node) with positively balanced Right Child -2 5 D 1 3 A 4 C B

40. AVL Tree - Double Rotations
Rotate Right (3) -2 5 D 1 3 A 4 C B

41. AVL Tree - Double Rotations
Update Balance (3) -2 5 D 4 C 1 3 B A

42. AVL Tree - Double Rotations
Update Balance (3) -2 5 D 4 C 3 B A

43. AVL Tree - Double Rotations
Rotate Left (5) -2
Reduced to Single Left (5) 5 D 1 4 C 3 B A

44. AVL Tree - Double Rotations
Update Balance (5) 1 4 -2 5 3 D C C A

45. AVL Tree - Double Rotations
Update Balance (5) 1 4 5 3 D C C A

46. AVL Tree - Double Rotations
Update Balance (4) 4 5 3 D C C A

47. AVL Tree - Quiz Insert 22. What will be the resulting tree? 17 25 9 5
TIME:
TIME’S UP!
Insert 22. What will be the resulting tree? 17 25 9 5 20 12

48. AVL Tree - Quiz   Insert 22. What will be the resulting tree? 17 25
TIME’S UP!
Insert 22. What will be the resulting tree? 17 25 9 5 20 12 22 17 25 9 5 12 20 17 25 9 5 12 22 20  

49. Insert/Delete perform O(lgN) rotations
AVL Tree - Summary
Structure
Worst case
Average case
Search
Insert
Delete
Search Hit
BST N
1.39 lg N
2-3 Tree
c lg N
Red-Black
2 lg N
lg N
AVL Tree
1.44 lg N
Insert/Delete perform O(lgN) rotations

50. Balancing Implementation22 17 25 9 5 12 20
AVL Tree
Balancing Implementation

51. 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; }

52. 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;

53. Lab Exercise
AVL Tree - Balancing

54. AA Tree Utilizes the concept of levels
Level - the number of left links on the path to a null node 17
Level 3 9 25
Level 2 5 12 22 32
Level 1

55. 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

56. AA Tree Right horizontal links are possible
Left horizontal links are not allowed 17
Level 3 9 25
Level 2 5 10 12 22 32 55
Level 1

57. Skew Skew operation is a single right rotation
Skew when an insertion or deletion creates a horizontal left link 17
Level 3 9 25
Level 2 5 10 12 22 32 55
Level 1

58. Skew (2) Skew operation is a single right rotation
Skew when an insertion or deletion creates a horizontal left link 17
Level 3 9 25
Level 2 5 10 12 22 32 55
Level 1

59. Split Split operation is a single left rotation
Split when an insertion or deletion two consecutive right horizontal links 17
Level 3 9 25
Level 2 5 10 12 22 32 55 71
Level 1

60. Split Split operation is a single left rotation
Split when an insertion or deletion two consecutive right horizontal links 17
Level 3 9 25 55
Level 2 5 10 12 22 32 71
Level 1

61. 17 25 9 5 12 22 32 55 10 71
AA Tree
Insertion Algorithm

62. AA Tree Insertion #1 Insert: 6
New nodes are always inserted at Level 1
Level 3
Level 2 6
Level 1

63. AA Tree Insertion #1
Insert: 2
Level 3
Level 2 6
Level 1

64. AA Tree Insertion #1 Left horizontal link is not allowed
Rotate right (skew) 6
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
Insert: 8
Level 3
Level 2 2 6
Level 1

68. AA Tree Insertion #1 Two consecutive right horizontal links
Rotate left (split) 2
Level 3
Level 2 2 6 8
Level 1

69. AA Tree Insertion #1
Insert: 16
Level 3 6
Level 2 2 8
Level 1

70. AA Tree Insertion #1
Insert: 16
Level 3 6
Level 2 2 8
Level 1

71. AA Tree Insertion #1
Insert: 16
Level 3 6
Level 2 2 8
Level 1

72. AA Tree Insertion #1
Insert: 16
Level 3 6
Level 2 2 8 16
Level 1

73. AA Tree Insertion #1
Insert: 10
Level 3 6
Level 2 2 8 16
Level 1

74. AA Tree Insertion #1
Insert: 10
Level 3 6
Level 2 2 8 16
Level 1

75. AA Tree Insertion #1
Insert: 10
Level 3 6
Level 2 2 8 16
Level 1

76. AA Tree Insertion #1 Horizontal left link not allowed
Rotate right (skew) 16
Level 3 6
Level 2 2 8 16
Level 1 10

77. AA Tree Insertion #1 Two consecutive right horizontal links
Rotate left (split) 8
Level 3 6
Level 2 2 8 10 16
Level 1

78. AA Tree Insertion #1 Two consecutive right horizontal links
Rotate left (split) 8
Level 3 6 10
Level 2 2 8 16
Level 1

79. Trees - Quiz Consider web application in which
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+

80. AVL trees are more rigidly balanced, so they have faster search
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
Red-Black B+
AVL trees are more rigidly balanced, so they have faster search

81. 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

82. AVL Trees, AA Trees
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"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
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