1. Tree traversals BST properties Search Insertion
Binary Search Trees
Tree traversals
BST properties
Search
Insertion
October 30, 2017
Hassan Khosravi / Geoffrey Tien

2. Hassan Khosravi / Geoffrey Tien
Binary tree traversal
A traversal algorithm for a binary tree visits each node in the tree
Typically, it will do something while visiting each node!
Traversal algorithms are naturally recursive
There are three traversal methods
inOrder
preOrder
postOrder
October 30, 2017
Hassan Khosravi / Geoffrey Tien

3. inOrder traversal algorithm
void inOrder(BNode* nd) {
if (nd != NULL)
inOrder(nd->left);
visit(nd);
inOrder(nd->right); }
typedef struct BNode {
int data;
struct BNode* left;
struct BNode* right;
} BNode;
The visit function would do whatever the purpose of the traversal is (e.g. print the data value of the node).
October 30, 2017
Hassan Khosravi / Geoffrey Tien

4. Hassan Khosravi / Geoffrey Tien
preOrder traversal
visit(nd);
preOrder(nd->left); 1
preOrder(nd->right); 17 2 6
visit
visit 13 27
preOrder(left)
preOrder(left)
preOrder(right)
preOrder(right) 3 5 7 8
visit
visit
visit 9 16 20 39
preOrder(left)
preOrder(left)
preOrder(left)
preOrder(right)
preOrder(right)
preOrder(right)
visit 4
preOrder(left)
preOrder(right) 11
visit
preOrder(left)
preOrder(right)
October 30, 2017
Hassan Khosravi / Geoffrey Tien

5. Hassan Khosravi / Geoffrey Tien
postOrder traversal
postOrder(nd->left);
postOrder(nd->right); 8
visit(nd); 17 4 7
postOrder(left)
postOrder(left) 13 27
postOrder(right)
postOrder(right)
visit
visit 2 3 5 6
postOrder(left)
postOrder(left)
postOrder(left) 9 16 20 39
postOrder(right)
postOrder(right)
postOrder(right)
visit
visit
visit
postOrder(left) 1
postOrder(right)
visit
postOrder(left) 11
postOrder(right)
visit
October 30, 2017
Hassan Khosravi / Geoffrey Tien

6. Hassan Khosravi / Geoffrey Tien
Exercise
What will be printed by an in-order traversal of the tree?
preOrder? postOrder?
inOrder(nd->left); 17
visit(nd);
inOrder(nd->right); 9 27 6 16 20 31
Food for thought: which traversal will be useful for copying a tree? Deallocating all nodes? 12 39
October 30, 2017
Hassan Khosravi / Geoffrey Tien

7. Before we begin...
Another type of tree traversal
We have seen pre-order, in-order, post-order traversals
What about a traversal that visits every node in a level before working on the next level?
level-order traversal 41 33 87 21 74 36 45 78 25
Use some ADT to support this?
October 30, 2017
Hassan Khosravi / Geoffrey Tien

8. Binary search trees
A data structure for the Dictionary ADT?
A binary search tree is a binary tree with a special property
For all nodes in the tree:
All nodes in a left subtree have labels less than the label of the subtree's root
All nodes in a right subtree have labels greater than or equal to the label of the subtree's root
Binary search trees are fully ordered
October 30, 2017
Hassan Khosravi / Geoffrey Tien

9. Hassan Khosravi / Geoffrey Tien
BST example 17 13 27 9 16 20 39 11
October 30, 2017
Hassan Khosravi / Geoffrey Tien

10. Hassan Khosravi / Geoffrey Tien
BST inOrder traversal
inOrder(nd->leftchild);
inOrder traversal on a BST retrieves data in sorted order
visit(nd); 5
inOrder(nd->rightchild); 17 3 7
inOrder(left)
inOrder(left) 13 27
visit
visit
inOrder(right)
inOrder(right) 1 4 6 8
inOrder(left)
inOrder(left)
inOrder(left) 9
visit 16 20
visit 39
visit
inOrder(right)
inOrder(right)
inOrder(right)
inOrder(left) 2
visit
inOrder(right)
inOrder(left) 11
visit
inOrder(right)
October 30, 2017
Hassan Khosravi / Geoffrey Tien

11. Hassan Khosravi / Geoffrey Tien
BST implementation
Binary search trees can be implemented using a reference structure
Tree nodes contain data and two pointers to nodes
BNode* leftchild
data
BNode* rightchild
Data to be stored in the tree (usually an object)
References or pointers to other tree Nodes
October 30, 2017
Hassan Khosravi / Geoffrey Tien

12. Hassan Khosravi / Geoffrey Tien
BST search
To find a value in a BST search from the root node:
If the target is less than the value in the node search its left subtree
If the target is greater than the value in the node search its right subtree
Otherwise return true, (or a pointer to the data, or …)
How many comparisons?
One for each node on the path
Worst case: height of the tree + 1
October 30, 2017
Hassan Khosravi / Geoffrey Tien

13. Hassan Khosravi / Geoffrey Tien
BST search example
search(27); 17 27
October 30, 2017
Hassan Khosravi / Geoffrey Tien

14. Hassan Khosravi / Geoffrey Tien
BST search example
search(16); 17 13 16
October 30, 2017
Hassan Khosravi / Geoffrey Tien

15. Hassan Khosravi / Geoffrey Tien
BST search example
search(12); 17 13 9 11
October 30, 2017
Hassan Khosravi / Geoffrey Tien

16. Search implementation
Search can be implemented iteratively or recursively
int search(BNode* nd, int key) {
if (nd == NULL) return FALSE;
else if (nd->data == key) return TRUE;
else {
if (key < nd->data)
return search(nd->left, key);
else
return search(nd->right, key); }
October 30, 2017
Hassan Khosravi / Geoffrey Tien

17. Hassan Khosravi / Geoffrey Tien
BST insertion
The BST property must hold after insertion
Therefore the new node must be inserted in the correct position
This position is found by performing a search
If the search ends at the (null) left child of a node make its left child refer to the new node
If the search ends at the right child of a node make its right child refer to the new node
The cost is about the same as the cost for the search algorithm, O(height)
October 30, 2017
Hassan Khosravi / Geoffrey Tien

18. Hassan Khosravi / Geoffrey Tien
BST insertion example
Insert 43 47
Create new node
Find position 32 63
Link node 19 41 54 79 43 10 23 37 44 53 59 96 7 12 30 43 57 91 97
October 30, 2017
Hassan Khosravi / Geoffrey Tien

19. Hassan Khosravi / Geoffrey Tien
BST insertion example
Create new BST 3
Insert 3
Insert 15 15
Insert 21
Insert 23 21
Insert 37
Search 45 23
How many operations for Search? Complexity? 37
October 30, 2017
Hassan Khosravi / Geoffrey Tien

20. Insert implementation
Insert can also be implemented iteratively or recursively
BNode* insert(BNode* nd, int key) {
if (nd == NULL) {
BNode* newnode = (BNode*) malloc(sizeof(BNode));
newnode->data = key;
newnode->left = NULL;
newnode->right = NULL;
return newnode; }
else {
if (key < nd->data)
nd->left = insert(nd->left, key);
else
nd->right = insert(nd->right, key);
return nd;
October 30, 2017
Hassan Khosravi / Geoffrey Tien

21. Hassan Khosravi / Geoffrey Tien
Find Min, Find Max
Find minimum:
From the root, keep following left child links until no more left child exists (i.e. NULL)
Find maximum:
From the root, follow right child links until no more right child exists 43 18 68 12 7 9 33 52 56 67 27 39 50 21
October 30, 2017
Hassan Khosravi / Geoffrey Tien

22. Hassan Khosravi / Geoffrey Tien
Exercise
Write iterative implementations for:
// returns TRUE or FALSE int search(BNode* nd, int key)
// returns the root of the subtree receiving the // inserted item BNode* insert(BNode* nd, int key)
// returns the value of the minimum key int findMin(BNode* nd)
// returns the value of the maximum key int findMax(BNode* nd)
October 30, 2017
Hassan Khosravi / Geoffrey Tien

23. Readings for this lesson
Thareja
Chapter 9.4
10.1 – , – , –
Next class:
Thareja Chapter
October 30, 2017
Hassan Khosravi / Geoffrey Tien