1. AVL Trees
"The voyage of discovery is not in seeking new landscapes but in having new eyes. "
- Marcel Proust

2. Binary Search Trees
All the operations on a binary search tree cost 𝑂(ℎ), where ℎ is the height of the tree.
These operations are efficient when ℎ= O(log 𝑛) .
However, it may happen that ℎ=𝑂(𝑛).
When the tree is entirely unbalanced.
The cost of any operation is linear in the number of nodes and not logarithmic.
Solution: Keep the tree balance to ensure operations always cost O(log n)!
CS Data Structures

3. Measuring Tree Balance
In a tree, the balance factor of a node x is defined as the difference in the heights of its two sub-trees:
𝐵𝑎𝑙𝑎𝑛𝑐𝑒𝐹𝑎𝑐𝑡𝑜𝑟 𝑥 =𝐻𝑒𝑖𝑔ℎ𝑡 𝑅𝑖𝑔ℎ𝑡 𝑥 −𝐻𝑒𝑖𝑔ℎ𝑡(𝐿𝑒𝑓𝑡 𝑥 )
Node height is the number of edges between a node and the furthest leaf in its sub-trees.
Nodes with balance factors <0 are called "left-heavy."
Nodes with balance factors >0 are called "right-heavy."
Nodes with balance factors =0 are called "balanced."
A balanced tree is a tree in which all of its nodes are balanced.
CS Data Structures

4. Calculating Balance Factor
For instance, the balance factor of J is:
𝐵𝑎𝑙𝑎𝑛𝑐𝑒𝐹𝑎𝑐𝑡𝑜𝑟 𝐽 =𝐻𝑒𝑖𝑔ℎ𝑡 𝑅𝑖𝑔ℎ𝑡 𝐽 −𝐻𝑒𝑖𝑔ℎ𝑡 𝐿𝑒𝑓𝑡 𝐽
𝐵𝑎𝑙𝑎𝑛𝑐𝑒𝐹𝑎𝑐𝑡𝑜𝑟 𝐽 =𝐻𝑒𝑖𝑔ℎ𝑡 𝑃 −𝐻𝑒𝑖𝑔ℎ𝑡 𝐹
𝐵𝑎𝑙𝑎𝑛𝑐𝑒𝐹𝑎𝑐𝑡𝑜𝑟 𝐽 =3−2
𝐵𝑎𝑙𝑎𝑛𝑐𝑒𝐹𝑎𝑐𝑡𝑜𝑟 𝐽 =+1
CS Data Structures

5. Balancing Trees Don’t balance Strict balance Good balance
May end up with some nodes very deep
Strict balance
The tree must always be balanced perfectly
Expensive
must ensure tree complete after every modification
Good balance
Allow some imbalance
Adjust on access
Self-adjusting
CS Data Structures

6. Balancing Binary Search Trees
Ignoring balance leads to poor performance
Maintaining perfect balance is too costly
Many algorithms exist for keeping good balance
Height-balanced trees
AVL trees (Adelson-Velskii and Landis)
Self-adjusting trees
Splay trees
Multiway search trees
B-Trees
CS Data Structures

7. AVL-Trees An AVL tree is a self-balancing, binary search tree.
named after its inventors, Adelson-Velskii and Landis.
first proposed dynamically balancing trees.
not perfectly balanced, but each node has a balance factor of -1, 0, or 1.
maintains 𝑂 log 𝑛 search, addition and deletion time.
CS Data Structures

8. Unbalanced AVL-Trees
Insertions and deletions may change the balance factor of nodes in a binary search tree
For instance, given this deletion: 6 -1 6 -2 3 8 3 +1
Delete 8 2 5 2 5 +1
CS Data Structures

9. Unbalanced AVL-Trees or this insertion: 6 6 3 8 3 8 Insert 4 2 5 2 5 4-1 6 -2 3 8 3 +1 8
Insert 4 2 5 2 5 +1 4
CS Data Structures

10. AVL-Tree Imbalance
After an insertion or a deletion, one or more nodes can become too far out of balance (i.e. have balance factors of −2 𝑜𝑟 2).
These imbalances can occur in four places:
outside
in the left sub-tree
in the right sub-tree
inside
in the right sub-tree of the left sub-tree
in the left sub-tree of the right sub-tree
CS Data Structures

11. AVL-Tree Imbalances Outside: -Left sub-tree -Right sub-tree
Note: T1, T2, and T3 represent arbitrary subtrees, which may be empty or may contain any number of nodes.
CS Data Structures

12. AVL-Tree Imbalances Inside: - Right sub-tree of left sub-tree
- Left sub-tree of right sub-tree
CS Data Structures

13. AVL-Tree Rotation
To rebalance an AVL-Tree, perform one or more tree rotations.
There are 4 types of tree rotations, depending on where the imbalance occurs in the tree:
Outside:
LL-Rotation – right sub-tree’s, right sub-tree imbalance
RR-Rotation – left sub-tree’s, left sub-tree imbalance
Inside:
LR-Rotation – left sub-tree’s, right sub-tree imbalance
a LL-Rotation followed by a RR-Rotation
RL-Rotation – right sub-tree’s, left-sub-tree imbalance
a RR-Rotation followed by a LL-Rotation
CS Data Structures

14. Key Idea
If a node is out of balance in a given direction, rotate it in the opposite direction.
rotation: A swap between a parent and its left or right child, maintaining proper BST ordering. 25 8
rotate right 8 3 25 3 11 11
CS Data Structures

15. RR-Rotations
When insert a node into the left sub-tree of a left sub-tree, it may cause a sub-tree imbalance.
To rebalance, perform a RR-Rotation in the clockwise direction. 𝑅𝑅
CS Data Structures

16. Example: RR-Rotation
Below, when A is inserted into the left sub-tree of C’s left sub-tree, C has a -2 imbalance.
We perform a RR-Rotation by making C the right-subtree of its left sub-tree, B.
Note: If B had a right sub-tree, that sub-tree would become C’s left sub-tree (i.e. a second right rotation). -2 -1
CS Data Structures

17. LL-Rotations
When insert a node into the right sub-tree of a right sub-tree, it may cause a sub-tree imbalance.
To rebalance the tree, perform a LL-Rotation, in the counter-clockwise direction. 𝐿𝐿
CS Data Structures

18. Example: LL-Rotation
Below, when C is inserted into the right sub-tree of A’s right sub-tree, A has a +2 imbalance.
Perform a left rotation by making A the left-subtree of its right sub-tree, B.
Note: If B had a left sub-tree, that sub-tree would become A’s right sub-tree (i.e a second left rotation). +2 +1
CS Data Structures

19. LR-Rotations
When insert a node into the right sub-tree of a left sub-tree, it may cause a sub-tree imbalance.
To rebalance, perform a LR-Rotation.
Inside rotations are combinations of the outside rotations.
A LR-Rotation is the combination of a LL-Rotation and a RR-Rotation. 𝐿𝑅
CS Data Structures

20. Example: LR-Rotation -2
For example, when B is inserted into the right sub-tree of C’s left sub-tree, C has a -2 imbalance. +1
Perform a LL-Rotation on the left sub-tree of C. This makes A the left sub-tree of B.
Note: If B had a left sub-tree, it would become A’s right sub-tree.
CS Data Structures

21. Example: LR-Rotation -2
The sub-tree is still unbalanced. However, it’s now because of the left sub-tree of the left sub-tree. -1
Perform a RR-Rotation, making B the new root node of this sub-tree. C becomes the right sub-tree of its left sub-tree, B.
Note: If B had a right sub-tree, it would become C’s left sub-tree.
The sub-tree is now balanced.
CS Data Structures

22. RL-Rotations
When insert a node into the left sub-tree of a right sub-tree, it may cause a sub-tree imbalance.
To rebalance, perform a RL-Rotation.
A RL-Rotation is the combination of a RR-Rotation a LL-Rotation. 𝑅𝐿
CS Data Structures

23. Example: RL-Rotation +2
For example, when B is inserted into the left sub-tree of A’s right sub-tree, B has a +2 imbalance. -1
Perform a RR-Rotation on the right sub-tree of A. This makes C the right sub-tree of B.
Note: If B had a right sub-tree, it would become C’s left sub-tree.
CS Data Structures

24. Example: RL-Rotation +2
The sub-tree is still unbalanced. However, it’s now because of the right sub-tree of the right sub-tree. -1
Perform LL-Rotation, making B the new root of the sub-tree. A becomes the left sub-tree of its right sub-tree B.
Note: If B had a left sub-tree, it would become A’s right sub-tree.
The sub-tree is now balanced.
CS Data Structures

25. AVL-Tree Insertion
Insert the new node into the tree, like insertion into a binary search tree
Fix tree balance via rotations, if necessary
A rotation is performed in sub-trees with a root that has a balance factor equal to +2 or -2.
If there is more than one imbalance, first rotate at the node closest to the newly inserted node.
CS Data Structures

26. Insert 15 in the following AVL-Tree:
Example: Insertion I
Insert 15 in the following AVL-Tree: 10 20 5 12 4 7 8 2 -1 +1
CS Data Structures

27. Insertion induces left, right imbalance
Example: Insertion I
Insertion induces left, right imbalance 10 20 5 12 4 7 8 2 -1 -2 +1 15
CS Data Structures

28. Example: Insertion I
Which rotation use?
CS Data Structures

29. After apply LR-Rotation at node 20
Example: Insertion I
After apply LR-Rotation at node 20 10 15 5 12 4 7 8 2 -1 +1 20
CS Data Structures

30. Insert 9 in the following AVL-Tree:
Example: Insertion II
Insert 9 in the following AVL-Tree: 10 20 5 12 4 7 8 2 -1 +1
CS Data Structures

31. Example: Insertion II
Insertion induces left, right imbalance 10 20 5 12 4 7 8 2 -2 -1 +1 +2
We rotate at 7, closest node to 9 9
and a right, right imbalance
CS Data Structures

32. Example: Insertion II
Which rotation use?
CS Data Structures

33. After apply LL-Rotation at node 7
Example: Insertion II 10 20 5 12 4 8 9 2 -1 7
After apply LL-Rotation at node 7
CS Data Structures

34. AVL-Trees Deletion Delete node as in other binary search trees
Fix tree balance via rotations, if necessary
A rotation is performed in sub-trees having a root that has a balance factor equal to +2 or -2.
If there is more than one imbalance, first rotate at the node closest to the deleted node.
CS Data Structures

35. Delete 7 from the following AVL-Tree:
Example: Deletion I
Delete 7 from the following AVL-Tree: 10 20 5 12 4 7 8 2 -1 +1
CS Data Structures

36. Example: Deletion I 10 20 5 12 4 8 2 No need to rotate! -1
No need to rotate!
After deleting 7
CS Data Structures

37. Delete 10 from the following AVL-Tree:
Example: Deletion II
Delete 10 from the following AVL-Tree: 10 20 5 12 4 7 8 2 -1 +1
CS Data Structures

38. Example: Deletion II
After deleting 10, promote its successor, 12
Could also promote its predecessor, 5 12 20 5 4 7 8 2 -2 -1 +1
CS Data Structures

39. Example: Deletion II
Deletion induces left, left imbalance
Choose to use RR-Rotation but could use a LR-Rotation too. 12 20 5 4 7 8 2 -2 -1 +1
and a left, right imbalance!
CS Data Structures

40. Example: Deletion II
Which rotation use?
CS Data Structures

41. After apply RR-Rotation at node 12
Example: Deletion II
After apply RR-Rotation at node 12 5 12 4 2 20 8 +1 -1 +1 7
CS Data Structures

42. Delete 4 from the following AVL-Tree:
Example: Deletion III
Delete 4 from the following AVL-Tree: 10 20 5 12 4 7 6 -1 +1
CS Data Structures

43. Deletion induces right, left imbalance
Example: Deletion III
Deletion induces right, left imbalance 10 20 5 12 7 6 -1 +2
CS Data Structures

44. Example: Deletion II
Which rotation use?
CS Data Structures

45. After apply RL-Rotation at node 5
Example: Deletion III 10 20 6 12 5 7 -1
After apply RL-Rotation at node 5
CS Data Structures

46. Height of AVL-Trees Maximum number of nodes in a full AVL-tree:
𝑛 ≤ 2 ℎ+1 −1
Then,
ℎ ≥ log 2 𝑛+1 −1 = Ω( log 𝑛 )
CS Data Structures

47. Minimum Number of Nodes
Minimum number of nodes in the AVL-tree:
nh: minimum number of nodes when the height is h
CS Data Structures

48. Minimum Number of Nodes
Minimum number of nodes in the AVL-tree:
Recall the Fibonacci numbers
nh = Fh + 1
CS Data Structures

49. Height of AVL-Trees Minimum number of nodes in the AVL-tree:
𝑛 ℎ = 𝐹 ℎ +1⟹ 𝑛 ℎ > 𝐹 ℎ
𝐹 ℎ is the closest integer to 𝜙 ℎ , where 𝜙=1.618
Then, 𝐹 ℎ ≈ 𝜙 ℎ 5
and 𝑛 ℎ > 𝜙 ℎ ,
then ℎ< log 𝜙 log 𝜙 𝑛 =𝑂( log 𝑛 )
Since h is Ω( log 𝑛 )and 𝑂 log 𝑛 , h is Θ( log 𝑛 ).
CS Data Structures

50. Disadvantages of AVL-Trees
Difficult to program and debug
Asymptotically faster than other self-adjusting trees but rebalancing takes time.
Most large searches are done in database systems on disk and use other structures (e.g. B-trees).
A 𝑂(𝑛) runtime for a single operation may be justified, if total run time for many consecutive operations is fast (e.g. Splay trees).
CS Data Structures

51. Advantages of AVL-Trees
Since AVL-Trees are always close to balanced, search is 𝑂( log 𝑛 ).
Insertion and deletion are also 𝑂( log 𝑛 ).
For both insertion and deletion, height re-balancing adds no more than a constant factor to the runtime.
CS Data Structures

52. CS Data Structures

53. AVL Insertion: Outside Casej
Consider a valid
AVL subtree k h Z h h X Y
CS Data Structures

54. AVL Insertion: Outside Casej
Inserting into X
destroys the AVL
property at node j k h Z
h+1 h Y X
CS Data Structures

55. AVL Insertion: Outside Casej
Do a “right rotation” k h Z
h+1 h Y X
CS Data Structures

56. j k Z Y X Single right rotation Do a “right rotation”
CS Data Structures

57. Outside Case Completedk
“Right rotation” done!
(“Left rotation” is mirror
symmetric) j
h+1 h h X Z Y
AVL property has been restored!
CS Data Structures

58. AVL Insertion: Inside Casej
Consider a valid
AVL subtree k h Z h h X Y
CS Data Structures

59. AVL Insertion: Inside Casej
Inserting into Y
destroys the
AVL property
at node j
Does “right rotation”
restore balance? k h Z h
h+1 X Y
CS Data Structures

60. AVL Insertion: Inside Casek
“Right rotation”
does not restore
balance… now k is
out of balance j h X h
h+1 Z Y
CS Data Structures

61. AVL Insertion: Inside Casej
Consider the structure
of subtree Y… k h Z h
h+1 X Y
CS Data Structures

62. AVL Insertion: Inside Casej
Y = node i and
subtrees V and W k h Z i h
h+1 X
h or h-1 V W
CS Data Structures

63. AVL Insertion: Inside Casej
We will do a left-right
“double rotation” . . . k Z i X V W
CS Data Structures

64. Double rotation : first rotationj
left rotation complete i Z k W V X
CS Data Structures

65. Double rotation : second rotationj
Now do a right rotation i Z k W V X
CS Data Structures

66. Double rotation : second rotation
right rotation complete
Balance has been
restored i k j h h
h or h-1 V X W Z
CS Data Structures

67. Implementation CS 321 - Data Structures balance (1,0,-1) key left
right
No need to keep the height; just the difference in height, i.e. the balance factor; this has to be modified on the path of insertion even if you don’t perform rotations
Once you have performed a rotation (single or double) you won’t need to go back up the tree
CS Data Structures