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

2. Which STL Container to Use?

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

4. 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

5. 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"

6. 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

7. 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

8. 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)

9. 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)

10. 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

11. 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

12. 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

13. 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

14. 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

15. 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

16. 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

17. 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

18. 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

19. 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

20. 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

21. 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

22. 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

23. 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

24. 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

25. Insert the following into an AVL tree: 14, 17, 11, 7, 53, 4, 13, 12, 8
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

26. Insert the following into an AVL tree: 14, 17, 11, 7, 53, 4, 13, 12, 8
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

27. Implementing an AVL Tree
Self-Balancing Search Trees

28. Performance of the AVL Tree
Self-Balancing Search Trees
Since each subtree is kept close to balanced, the AVL has expected O(log n).
Each subtree is allowed to be out of balance ±1 so the tree may contain some holes.
In the worst case (which is rare) an AVL tree can be 1.44 times the height of a full binary tree that contains the same number of items.
Ignoring constants, this still yields O(log n) performance.
Empirical tests show that on average log2.n comparisons are required to insert the nth item into an AVL tree – close to insertion into a corresponding complete binary search tree.

29. Pros and Cons of AVL Trees
Self-Balancing Search Trees

30. Pros and Cons of AVL Trees
Self-Balancing Search Trees
Argument for AVL trees:
Search is O(log n) since AVL trees are always balanced.
Insertion and deletions are also O(log n).
The height balancing adds no more than a constant factor to the speed of insertion.
Arguments against using AVL trees:
Difficult to program & debug; more space for balance factor.
Asymptotically faster but rebalancing costs time.
Most large searches are done in database systems on disk and use other structures (e.g. B-trees).
May be OK to have O(n) for a single operation if total run time for many consecutive operations is fast (e.g. Splay trees).