1. Trees

2. Recursion Review
Base Case- how we exit and don’t go into infinite recursion
Some work
Division of work
Recursive call (s)
For many of you, recursion is intimidating. Eventually it will become second nature just as loops and file operations have.

3. Recursive Process Decide if you want to use recursion
Decide inputs and outputs
Decide how to break down the work
Tentatively decide on a base case
Make recursive calls
Bring back together the results
Return the proper values
Finalize your base case
Trace the recursion
Practice sum numbers from 1 to n

4. Practice - Sum the numbers from 1 to n
int sum(int n){ if(n==1){ return n; } int curSum=n+sum(n-1); return curSum;

5. Practice - Sum array
int sum( int* arr, int pos, int size){ if(pos==size-1) return arr[pos]; return arr[pos]+sum(arr, pos+1, size); }

6. Practice - Linked List class Node { public: int value; Node *next;
Node( int val, Node *n = NULL ){
value=val;
next=n; } };

7. Practice - Linked List Print
Void print(Node* cur){ if( cur==NULL) return; cout<<cur->value; print(cur->next); }

8. Binary Search Tree What is a tree? What is a Binary tree?
What is a Binary search tree?
Binary tree, node and two pointers to children
Binary search tree is a binary tree
But it is ordered to facilitate searching
Anything to the left is a smaller value, anything to the right is a greater value
Children are binary search trees

9. Binary Search Tree
Is a binary tree (each node has two pointers to children)
Ordered to facilitate searching
Anything to the left is a smaller value
Anything to the right is a greater value
Children are binary search trees
Cannot have duplicate values because it violates the search property

10. Example 1

11. At your seats
Build a tree from these nodes
B R L M T C A N P D

12. Deletion We need to preserve the binary search properties
Deleting a leaf is easy, we can just delete it, and set its parent’s pointer to null
The easiest way to delete a node is to find its successor, copy the value over, then delete that leaf node

13. Code - Node
class Node { public: int value; Node *left; Node *right; Node( int val, Node *l = NULL, Node *r = NULL ){ value=val; left=l; right=r; } };

14. Code - Search
Node* search( int value, Node* cur){ if(cur==NULL) return NULL; else if( value == cur->value) return cur; else if( value < cur->value) return search(value, cur->left); else return search(value, cur->right); }

15. Code - Delete
void delete( int value, Node* cur){ }

16. Complexity Search Insert Delete
Search – Ο(h) , where h is the height of the tree.
In the best case (balanced), h is log n . In the worst case, h is n (the number of nodes in the tree).
Insert – Ο(h) , where h is the height of the tree.
Delete – Ο(h) , where h is the height of the tree.

17. Disadvantages What is my worst case? When will it occur? O(n)
Will happen if things are in order

18. Balanced
Balanced means the heights of the left and right sub-trees differ by at most one
This needs to be true for every node in the tree
If a tree is balanced, what is the worst case?

19. Balanced Examples

20. Alternatives
AVL and Splay trees
Self balancing

21. AVL - Adelson, Velskii, Landis
Binary tree
Balanced
Determining if it is an AVL tree
Check each node for
Right sub-tree is an AVL tree
Left sub-tree is an AVL tree
Heights of right and left differ by at most one
AVL search Tree
An AVL tree, but also a binary tree

22. AVL practice
Is it AVL? Is it AVL search?

23. AVL We usually only look at AVL search
When I say AVL, I mean AVL search
If not search, I will specify by saying non-search AVL

24. Properties Height is logn Can do it for any n Search is logn
Insertion is logn
Deletion is logn

25. AVL - Search
How do we search?
Same as binary search

26. AVL - Insert How is it done? Same way as binary search
What if it is thrown off balance?
Re- balance
Rotations

27. AVL - Rotations RR LL RL LR

28. AVL - Single Left Rotation
Fixes a RR inbalance

29. AVL - Double Left Rotation
Fixes an RL inbalance

30. AVL Insert
3, 2, 1, 4, 5, 6, 7, 16, 15, 14

31. AVL Delete
How do we delete?

32. AVL Delete How do we delete? Same way as in a binary search tree
But the tree can be thrown off balance
Re- balance the tree using rotations
May need to do multiple rotations.

33. AVL Delete Rebalancing
Starting where we deleted, we follow the recursion back up the tree
At every node we visit, we need to check that it is still balanced.
If it is not balanced, we use a rotation to fix it
What is the worst case for the number of rotations I have to do?

34. AVL Delete
Delete 60

35. More AVL Delete
Delete 70

36. Example - Delete

37. Example - Delete

38. Example - Delete

39. Delete – Code?
int Delete (){ }

40. Participation Quiz – 5 points
Do you like the way I use power points to guide the lecture? Suggestions?
Do you feel I do a good job answering in class questions? Suggestions?
Do you find the examples, and asking for class guidance through them helpful? Suggestions?
Write a short snipet of code that has O(n) complexity. It can be simple, and does not need to do anything useful.
Which trees drawn on the board are AVL? (remember I mean AVL search)
I want this to be the best learning experience possible for you, so please include any other comments or suggestions you have for improving the course.