1. Self-Balancing Search Trees
Chapter 11
Self-Balancing Search Trees
Chapter 11

2. Tip #38: Very Late?
Self-Balancing Search Trees
Is someone’s junior year in college very late to learn programming?
Yes, give up…
Yes, give up… Or,
You can take the advice of this wise Chinese proverb:
“The best time to plant a tree was 20 years ago.
The second best time is now.”
You are still young. You have most of your life ahead of you.
Regardless of what field or trade you pick up, you WILL have to learn something new, whether you like it or not.
So, why not learn something that interests you?

3. Self-Balancing Search Trees
Chapter 11
Self-Balancing Search Trees
11.1, pgs

4. Tree Terminology Summary
Self-Balancing Search Trees
Branch (Edge)
Node (Vertex)
Depth
Height: 4
Depth: 0
Level: 1
Height
Height: 2
Depth: 1
Level: 2
Height: 3
Depth: 1
Level: 2
Height: 1
Depth: 2
Level: 3
Height: 1
Depth: 2
Level: 3
Height: 2
Depth: 2
Level: 3
Height: 1
Depth: 3
Level: 4
Root Node
Internal Node
Leaf Node
Root node is depth 0
Leaf nodes are height 1

5. Self-Balancing Search Trees
The performance of a binary search tree is proportional to the height of the tree or the maximum number of nodes along a path from the root to a leaf.
A full binary tree of height k can hold 2k - 1 items.
If a binary search tree is full and contains n items, the expected performance is O(log n).
However, if a binary tree is not full, the actual performance is worse than expected.
To solve this problem, we introduce self-balancing trees to achieve a balance so that the heights of the right and left subtrees are equal (or nearly equal.)
Self-balancing trees include the AVL binary search tree, and non-binary search trees such as the B-tree and its specializations, the 2-3 and trees, and the B+ tree.

6. 11.1, pgs. 624-628 11.1 Tree Balance and Rotation
Why Balance Is Important
Rotation
Algorithm for Rotation
Implementing Rotation
11.1, pgs

7. Why Balance is Important
Self-Balancing Search Trees
Searches into this unbalanced search tree are O(n), not O(log n).
A realistic example of an unbalanced tree.

8. Rotation
Self-Balancing Search Trees
A binary tree with n nodes (root, leaf, and internal nodes) and of height h is balanced if:
2(h − 1) ≤ n < 2h
Otherwise, it is unbalanced.
For example, a binary tree with height 4 can have between 8 and 15 nodes (between 1 and 8 leaf nodes) to be balanced.
We need an operation on a binary tree that changes the relative heights of left and right subtrees, but preserves the binary search tree property.

9. Algorithm for Right Rotation
Self-Balancing Search Trees 20 10 40 5 15 7
root
root
= left
right =
data = 20
BTNode
= left
right =
data = 10
BTNode
= left
right =
data = 40
NULL
BTNode
= left
right =
data = 5
NULL
BTNode
= left
right =
data = 15
NULL
BTNode
Right rotate around 20
= left
right =
data = 7
NULL
BTNode

10. Algorithm for Right Rotation
Self-Balancing Search Trees 20 10 40 5 15 7
root
root
= left
right =
data = 20
BTNode
temp
temp
= left
right =
data = 10
BTNode
= left
right =
data = 40
NULL
BTNode
= left
right =
data = 5
NULL
BTNode
= left
right =
data = 15
NULL
BTNode
Remember value of root->left (temp = root->left)
= left
right =
data = 7
NULL
BTNode

11. Algorithm for Right Rotation
Self-Balancing Search Trees 20 10 40 5 15 7
root
temp
root
= left
right =
data = 20
BTNode
temp
= left
right =
data = 10
BTNode
= left
right =
data = 40
NULL
BTNode
= left
right =
data = 5
NULL
BTNode
= left
right =
data = 15
NULL
BTNode
Remember value of root->left (temp = root->left)
Set root->left to value of temp->right
= left
right =
data = 7
NULL
BTNode

12. Algorithm for Right Rotation
Self-Balancing Search Trees 20 10 40 5 15 7
root
temp
root
= left
right =
data = 20
BTNode
temp
= left
right =
data = 10
BTNode
= left
right =
data = 40
NULL
BTNode
= left
right =
data = 5
NULL
BTNode
= left
right =
data = 15
NULL
BTNode
Remember value of root->left (temp = root->left)
Set root->left to value of temp->right
Set temp->right to root
= left
right =
data = 7
NULL
BTNode

13. Algorithm for Right Rotation
Self-Balancing Search Trees 20 10 40 5 15 7
root
temp
root
= left
right =
data = 20
BTNode
temp
= left
right =
data = 10
BTNode
= left
right =
data = 40
NULL
BTNode
= left
right =
data = 5
NULL
BTNode
= left
right =
data = 15
NULL
BTNode
Remember value of root->left (temp = root->left)
Set root->left to value of temp->right
Set temp->right to root
Set root to temp
= left
right =
data = 7
NULL
BTNode

14. Algorithm for Right Rotation
Self-Balancing Search Trees
= left
right =
data = 20
BTNode
data = 10
data = 40
NULL
data = 5
data = 15
data = 7
root
Remember value of root->left (temp = root->left)
Set root->left to value of temp->right
Set temp->right to root
Set root to temp 20 10 40 5 15 7
root 10 5 20 15 7 40
root

15. 11.2 AVL Trees 11.2, pgs. 628-643 Balancing a Left-Left Tree
Balancing a Left-Right Tree
Four Kinds of Critically Unbalanced Trees
Implementing an AVL Tree
Inserting into an AVL Tree
Removal from an AVL Tree
Performance of the AVL
11.2, pgs

16. AVL Trees
Self-Balancing Search Trees
In 1962 G.M. Adel'son-Vel'skiî and E.M. Landis developed a self-balancing tree. The tree is known by their initials: AVL.
The AVL tree algorithm keeps track of the difference in height of each subtree.
As items are added to or removed from a tree, the balance of each subtree from the insertion or removal point up to the root is updated.
BalanceFactor = height(right-subtree) – height(left-subtree)
The absolute difference between the left sub tree and right sub tree is never greater than 1.
If the balance gets out of the range -1 to +1, the tree is rotated to bring it back into range.

17. Balanced: 2(h − 1) ≤ n < 2h AVL: |Rh – Lh| ≤ 1
Q38.1: Which of these trees are 1) binary, 2) balanced, and 3) AVL trees? Why or why not? 20 10 40 5 15 7
Balanced: 2(h − 1) ≤ n < 2h
AVL: |Rh – Lh| ≤ 1

18. Not BST, not AVL, Balanced
Q38.1: Which of these trees are 1) binary, 2) balanced, and 3) AVL trees? Why or why not?
Not BST, not AVL, Balanced
(3,1)
(2,1)
(0,1)
BST, not AVL, not Balanced 20 10 40 5 15 7
Balanced: 2(h − 1) ≤ n < 2h
AVL: |Rh – Lh| ≤ 1
(3,2)
(0,1)
(1,2)
(1,1)
BST, AVL, Balanced
(3,3)
(0,2)
(1,2)
(1,1)
(0,1)
BST, not AVL, Balanced

19. Balancing a Left-Left Tree
Self-Balancing Search Trees
The dark purple trapezoid represents an insertion into this tree, making its height k + 1 50 25 c a
Each light purple triangle represents a tree of height k b

20. Balancing a Left-Left Tree
Self-Balancing Search Trees
The heights of the left and right subtrees are unimportant; only the relative difference matters when balancing 50
k - (k + 2) -2 25 c -1
k - (k + 1) a b
The formula hR – hL is used to calculate the balance of each node

21. Balancing a Left-Left Tree
Self-Balancing Search Trees
When the root and left subtree are both left-heavy, the tree is called a Left-Left tree 50 -2 25 c -1 a b
A Left-Left tree can be balanced by a rotation right

22. Balancing a Left-Left Tree
Self-Balancing Search Trees 25 a 50 b c
Even after insertion, the overall height has not increased

23. Balancing a Left-Right Tree
Self-Balancing Search Trees
k - (k + 2) 50 -2 25 c +1
(k + 1) - k a b
A Left-Right tree cannot be balanced by a simple rotation right

24. Balancing a Left-Right Tree
Self-Balancing Search Trees 50 -2 25 c +1 a b
Subtree b needs to be expanded into its subtrees bL and bR

25. Balancing a Left-Right Tree
Self-Balancing Search Trees 50 -2 25 c +1 40 a -1 bL bR
40 is left-heavy. The left subtree now can be rotated left

26. Balancing a Left-Right Tree
Self-Balancing Search Trees 50 -2 40 c -2 25 bR
The overall tree is now Left-Left and a rotation right will balance it. a bL

27. Balancing a Left-Right Tree
Self-Balancing Search Trees 40 50 25 +1 a bL bR c

28. 4 Critically Unbalanced Trees
Self-Balancing Search Trees
Left-Left (parent balance is -2, left child balance is -1)
Rotate right around parent
Left-Right (parent balance -2, left child balance +1)
Rotate left around child
Right-Right (parent balance +2, right child balance +1)
Rotate left around parent
Right-Left (parent balance +2, right child balance -1)
Rotate right around child

29. AVL Tree Example Build an AVL tree from the words:
Self-Balancing Search Trees
Build an AVL tree from the words:
"The quick brown fox jumps over the lazy dog"

30. AVL Tree Example The Insert The Self-Balancing Search Trees
Left-Left (parent balance is -2, left child balance is -1)
Rotate right around parent
Left-Right (parent balance -2, left child balance +1)
Rotate left around child
Right-Right (parent balance +2, right child balance +1)
Rotate left around parent
Right-Left (parent balance +2, right child balance -1)
Rotate right around child
The
Insert The

31. AVL Tree Example The +1 quick Insert quick Self-Balancing Search Trees
Left-Left (parent balance is -2, left child balance is -1)
Rotate right around parent
Left-Right (parent balance -2, left child balance +1)
Rotate left around child
Right-Right (parent balance +2, right child balance +1)
Rotate left around parent
Right-Left (parent balance +2, right child balance -1)
Rotate right around child
The +1
Insert quick
quick

32. AVL Tree Example The +2 -1 quick brown Insert brown
Self-Balancing Search Trees
Left-Left (parent balance is -2, left child balance is -1)
Rotate right around parent
Left-Right (parent balance -2, left child balance +1)
Rotate left around child
Right-Right (parent balance +2, right child balance +1)
Rotate left around parent
Right-Left (parent balance +2, right child balance -1)
Rotate right around child
The +2 -1
Insert brown
quick
brown
The overall tree is right-heavy (Right-Left)
1. Rotate right around the child (quick)

33. AVL Tree Example The brown +2 +1 quick
Self-Balancing Search Trees
Left-Left (parent balance is -2, left child balance is -1)
Rotate right around parent
Left-Right (parent balance -2, left child balance +1)
Rotate left around child
Right-Right (parent balance +2, right child balance +1)
Rotate left around parent
Right-Left (parent balance +2, right child balance -1)
Rotate right around child
The
brown +2 +1
quick
Rotate right around the child (quick)
Rotate left around the parent (The)

34. AVL Tree Example brown quick The Insert fox
Self-Balancing Search Trees
Left-Left (parent balance is -2, left child balance is -1)
Rotate right around parent
Left-Right (parent balance -2, left child balance +1)
Rotate left around child
Right-Right (parent balance +2, right child balance +1)
Rotate left around parent
Right-Left (parent balance +2, right child balance -1)
Rotate right around child
brown
quick
The
Insert fox

35. AVL Tree Example +1 brown The quick fox Insert fox
Self-Balancing Search Trees
Left-Left (parent balance is -2, left child balance is -1)
Rotate right around parent
Left-Right (parent balance -2, left child balance +1)
Rotate left around child
Right-Right (parent balance +2, right child balance +1)
Rotate left around parent
Right-Left (parent balance +2, right child balance -1)
Rotate right around child +1
brown
Insert fox
The
quick
fox

36. AVL Tree Example brown quick The +1 fox Insert jumps
Self-Balancing Search Trees
Left-Left (parent balance is -2, left child balance is -1)
Rotate right around parent
Left-Right (parent balance -2, left child balance +1)
Rotate left around child
Right-Right (parent balance +2, right child balance +1)
Rotate left around parent
Right-Left (parent balance +2, right child balance -1)
Rotate right around child
brown
quick
The +1
Insert jumps
fox

37. The tree is now left-heavy about quick (Left-Right case)
AVL Tree Example
Self-Balancing Search Trees
Left-Left (parent balance is -2, left child balance is -1)
Rotate right around parent
Left-Right (parent balance -2, left child balance +1)
Rotate left around child
Right-Right (parent balance +2, right child balance +1)
Rotate left around parent
Right-Left (parent balance +2, right child balance -1)
Rotate right around child +2 -2 +1
brown
Insert jumps
The
quick
fox
The tree is now left-heavy about quick (Left-Right case)
jumps

38. AVL Tree Example brown quick The +2 -2 fox +1 jumps
Self-Balancing Search Trees
Left-Left (parent balance is -2, left child balance is -1)
Rotate right around parent
Left-Right (parent balance -2, left child balance +1)
Rotate left around child
Right-Right (parent balance +2, right child balance +1)
Rotate left around parent
Right-Left (parent balance +2, right child balance -1)
Rotate right around child
brown
quick
The +2 -2
fox +1
Rotate left around the child (fox)
jumps

39. AVL Tree Example brown quick The +2 -2 jumps -1 fox
Self-Balancing Search Trees
Left-Left (parent balance is -2, left child balance is -1)
Rotate right around parent
Left-Right (parent balance -2, left child balance +1)
Rotate left around child
Right-Right (parent balance +2, right child balance +1)
Rotate left around parent
Right-Left (parent balance +2, right child balance -1)
Rotate right around child
brown
quick
The +2 -2
jumps -1
Rotate left around the child
fox
Rotate right around the parent

40. AVL Tree Example brown jumps The +1 fox quick
Self-Balancing Search Trees
Left-Left (parent balance is -2, left child balance is -1)
Rotate right around parent
Left-Right (parent balance -2, left child balance +1)
Rotate left around child
Right-Right (parent balance +2, right child balance +1)
Rotate left around parent
Right-Left (parent balance +2, right child balance -1)
Rotate right around child
brown
jumps
The +1
fox
quick

41. We now have a Right-Right imbalance
AVL Tree Example
Self-Balancing Search Trees
Left-Left (parent balance is -2, left child balance is -1)
Rotate right around parent
Left-Right (parent balance -2, left child balance +1)
Rotate left around child
Right-Right (parent balance +2, right child balance +1)
Rotate left around parent
Right-Left (parent balance +2, right child balance -1)
Rotate right around child +2 +1 -1
brown
Insert over
The
jumps
fox
quick
We now have a Right-Right imbalance
over

42. 1. Rotate left around the parent
AVL Tree Example
Self-Balancing Search Trees
Left-Left (parent balance is -2, left child balance is -1)
Rotate right around parent
Left-Right (parent balance -2, left child balance +1)
Rotate left around child
Right-Right (parent balance +2, right child balance +1)
Rotate left around parent
Right-Left (parent balance +2, right child balance -1)
Rotate right around child
brown
jumps
The +2 +1
fox
quick -1
1. Rotate left around the parent
over

43. AVL Tree Example jumps quick brown -1 The fox over Insert the
Self-Balancing Search Trees
Left-Left (parent balance is -2, left child balance is -1)
Rotate right around parent
Left-Right (parent balance -2, left child balance +1)
Rotate left around child
Right-Right (parent balance +2, right child balance +1)
Rotate left around parent
Right-Left (parent balance +2, right child balance -1)
Rotate right around child
jumps
quick
brown -1
The
fox
over
Insert the

44. AVL Tree Example jumps brown quick The fox over the Insert the
Self-Balancing Search Trees
Left-Left (parent balance is -2, left child balance is -1)
Rotate right around parent
Left-Right (parent balance -2, left child balance +1)
Rotate left around child
Right-Right (parent balance +2, right child balance +1)
Rotate left around parent
Right-Left (parent balance +2, right child balance -1)
Rotate right around child
jumps
Insert the
brown
quick
The
fox
over
the

45. AVL Tree Example jumps quick brown The fox over the Insert lazy
Self-Balancing Search Trees
Left-Left (parent balance is -2, left child balance is -1)
Rotate right around parent
Left-Right (parent balance -2, left child balance +1)
Rotate left around child
Right-Right (parent balance +2, right child balance +1)
Rotate left around parent
Right-Left (parent balance +2, right child balance -1)
Rotate right around child
jumps
quick
brown
The
fox
over
the
Insert lazy

46. AVL Tree Example +1 -1 jumps brown quick The fox over the lazy
Self-Balancing Search Trees
Left-Left (parent balance is -2, left child balance is -1)
Rotate right around parent
Left-Right (parent balance -2, left child balance +1)
Rotate left around child
Right-Right (parent balance +2, right child balance +1)
Rotate left around parent
Right-Left (parent balance +2, right child balance -1)
Rotate right around child +1 -1
jumps
Insert lazy
brown
quick
The
fox
over
the
lazy

47. AVL Tree Example jumps quick brown +1 -1 The fox over the lazy
Self-Balancing Search Trees
Left-Left (parent balance is -2, left child balance is -1)
Rotate right around parent
Left-Right (parent balance -2, left child balance +1)
Rotate left around child
Right-Right (parent balance +2, right child balance +1)
Rotate left around parent
Right-Left (parent balance +2, right child balance -1)
Rotate right around child
jumps
quick
brown +1 -1
The
fox
over
the
lazy
Insert dog

48. AVL Tree Example -1 +1 jumps brown quick The fox over the dog lazy
Self-Balancing Search Trees
Left-Left (parent balance is -2, left child balance is -1)
Rotate right around parent
Left-Right (parent balance -2, left child balance +1)
Rotate left around child
Right-Right (parent balance +2, right child balance +1)
Rotate left around parent
Right-Left (parent balance +2, right child balance -1)
Rotate right around child -1 +1
jumps
Insert dog
brown
quick
The
fox
over
the
dog
lazy