1. Operations on Binary Tree
There are a number of operations that can be defined for a binary tree.
If p is pointing to a node in an existing tree then
left(p) returns pointer to the left subtree
right(p) returns pointer to right subtree
parent(p) returns the father of p
brother(p) returns brother of p.
info(p) returns content of the node.
End of lecture 11

2. Operations on Binary Tree
In order to construct a binary tree, the following can be useful:
setLeft(p,x) creates the left child node of p. The child node contains the info ‘x’.
setRight(p,x) creates the right child node of p. The child node contains the info ‘x’.

3. Applications of Binary Trees
A binary tree is a useful data structure when two-way decisions must be made at each point in a process.
For example, suppose we wanted to find all duplicates in a list of numbers: 14, 15, 4, 9, 7, 18, 3, 5, 16, 4, 20, 17, 9, 14, 5

4. Applications of Binary Trees
One way of finding duplicates is to compare each number with all those that precede it.
14, 15, 4, 9, 7, 18, 3, 5, 16, 4, 20, 17, 9, 14, 5
14, 15, 4, 9, 7, 18, 3, 5, 16, 4, 20, 17, 9, 14, 5

5. Searching for Duplicates
If the list of numbers is large and is growing, this procedure involves a large number of comparisons.
A linked list could handle the growth but the comparisons would still be large.
The number of comparisons can be drastically reduced by using a binary tree.
The tree grows dynamically like the linked list.

6. Searching for Duplicates
The binary tree is built in a special way.
The first number in the list is placed in a node that is designated as the root of a binary tree.
Initially, both left and right subtrees of the root are empty.
We take the next number and compare it with the number placed in the root.
If it is the same then we have a duplicate.

7. Searching for Duplicates
Otherwise, we create a new tree node and put the new number in it.
The new node is made the left child of the root node if the second number is less than the one in the root.
The new node is made the right child if the number is greater than the one in the root.

8. Searching for Duplicates14
14, 15, 4, 9, 7, 18, 3, 5, 16, 4, 20, 17, 9, 14, 5

9. Searching for Duplicates15 14
15, 4, 9, 7, 18, 3, 5, 16, 4, 20, 17, 9, 14, 5

10. Searching for Duplicates14 15
15, 4, 9, 7, 18, 3, 5, 16, 4, 20, 17, 9, 14, 5

11. Searching for Duplicates14 4 15
4, 9, 7, 18, 3, 5, 16, 4, 20, 17, 9, 14, 5

12. Searching for Duplicates14 4 15
4, 9, 7, 18, 3, 5, 16, 4, 20, 17, 9, 14, 5

13. Searching for Duplicates14 9 4 15
9, 7, 18, 3, 5, 16, 4, 20, 17, 9, 14, 5

14. Searching for Duplicates14 4 15 9
9, 7, 18, 3, 5, 16, 4, 20, 17, 9, 14, 5

15. Searching for Duplicates7 14 4 15 9
7, 18, 3, 5, 16, 4, 20, 17, 9, 14, 5

16. Searching for Duplicates14 4 15 9 7
7, 18, 3, 5, 16, 4, 20, 17, 9, 14, 5

17. Searching for Duplicates18 14 4 15 9 7
18, 3, 5, 16, 4, 20, 17, 9, 14, 5

18. Searching for Duplicates14 4 15 9 18 7
18, 3, 5, 16, 4, 20, 17, 9, 14, 5

19. Searching for Duplicates3 14 4 15 9 18 7
3, 5, 16, 4, 20, 17, 9, 14, 5

20. Searching for Duplicates14 4 15 3 9 18 7
3, 5, 16, 4, 20, 17, 9, 14, 5

21. Searching for Duplicates5 14 4 15 3 9 18 7
5, 16, 4, 20, 17, 9, 14, 5

22. Searching for Duplicates14 4 15 3 9 18 7 5
5, 16, 4, 20, 17, 9, 14, 5

23. Searching for Duplicates16 14 4 15 3 9 18 7 5
16, 4, 20, 17, 9, 14, 5

24. Searching for Duplicates14 4 15 3 9 18 7 16 5
16, 4, 20, 17, 9, 14, 5

25. Searching for Duplicates4 14 4 15 3 9 18 7 16 5
4, 20, 17, 9, 14, 5

26. Searching for Duplicates20 14 4 15 3 9 18 7 16 5
20, 17, 9, 14, 5

27. Searching for Duplicates14 4 15 3 9 18 7 16 20 5
20, 17, 9, 14, 5

28. Searching for Duplicates17 14 4 15 3 9 18 7 16 20 5
17, 9, 14, 5

29. Searching for Duplicates14 4 15 3 9 18 7 16 20 5 17
17, 9, 14, 5

30. Searching for Duplicates14 4 15 3 9 18 7 16 20 5 17
9, 14, 5

31. C++ Implementation #include <stdlib.h>
template <class Object>
class TreeNode {
public:
// constructors
TreeNode() {
this->object = NULL;
this->left = this->right = NULL; };
TreeNode( Object* object )
this->object = object;

32. C++ Implementation Object* getInfo() { return this->object; };
void setInfo(Object* object)
this->object = object;
TreeNode* getLeft()
return left;
void setLeft(TreeNode *left)
this->left = left;

33. C++ Implementation TreeNode *getRight() { return right; };
void setRight(TreeNode *right)
this->right = right;
int isLeaf( )
if( this->left == NULL && this->right == NULL )
return 1;
return 0;

34. C++ Implementation private: Object* object; TreeNode* left;
TreeNode* right;
}; // end class TreeNode

35. C++ Implementation #include <iostream> #include <stdlib.h>
#include "TreeNode.cpp"
int main(int argc, char *argv[]) {
int x[] = { 14, 15, 4, 9, 7, 18, 3, 5, 16,4, 20, 17,
9, 14,5, -1};
TreeNode<int>* root = new TreeNode<int>();
root->setInfo( &x[0] );
for(int i=1; x[i] > 0; i++ )
insert(root, &x[i] ); }

36. C++ Implementation void insert(TreeNode<int>* root, int* info) {
TreeNode<int>* node = new TreeNode<int>(info);
TreeNode<int> *p, *q;
p = q = root;
while( *info != *(p->getInfo()) && q != NULL )
p = q;
if( *info < *(p->getInfo()) )
q = p->getLeft();
else
q = p->getRight(); }

37. C++ Implementation if( *info == *(p->getInfo()) ){
cout << "attempt to insert duplicate: "
<< *info << endl;
delete node; }
else if( *info < *(p->getInfo()) )
p->setLeft( node );
else
p->setRight( node );
} // end of insert

38. Trace of insert p 17 q 14 4 15 3 9 18 7 16 20 5
17, 9, 14, 5

39. Trace of insert p 17 14 4 15 q 3 9 18 7 16 20 5
17, 9, 14, 5

40. Trace of insert 17 14 p 4 15 q 3 9 18 7 16 20 5
17, 9, 14, 5

41. Trace of insert 17 14 p 4 15 3 9 q 18 7 16 20 5
17, 9, 14, 5

42. Trace of insert 17 14 4 15 p 3 9 18 q 7 16 20 5
17, 9, 14, 5

43. Trace of insert 17 14 4 15 p 3 9 18 7 q 16 20 5
17, 9, 14, 5

44. Trace of insert 17 14 4 15 3 9 18 p 7 16 20 q 5
17, 9, 14, 5

45. Trace of insert 17 14 4 15 3 9 18 p 7 16 20 q 5
17, 9, 14, 5

46. p->setRight( node );
Trace of insert 14 4 15 3 9 18 7 p 16 20
End of lecture 12 5 17
node
17, 9, 14, 5
p->setRight( node );