1. CSC 172– Data Structures and Algorithms
Fall 2017
TuTh 3:25 pm – 4:40 pm
Aug 30- Dec 22
Hoyt Auditorium

2. Announcement Sorting dance Extra credit opportunity:
(Any sorting algorithm)
Will replace your worst project score
You need to have 10+ students
Multiple workshop teams together?!
You have to upload the video to YouTube and provide us the link by Dec 08 (include the list of students participated)
Bonus points for the best dance!
CSC172, Fall 2017

3. Binary Trees Extremely useful data structure Special cases include
Huffman tree
Expression tree
Decision tree (in machine learning)
Heap data structure (later lecture)
Binary Trees
CSC172, Fall 2017

4. Binary Trees 5 Root left right 4 2 Depth 2 right 3 9 Height 39
Height 3
3, 7, 1, 9 are leaves 8 1
5, 4, 0, 8, 2 are internal nodes
Height 1 7
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5. Ancestors and Descendants5 4 2 3 9
1, 0, 4, 5 are ancestors of 1 8 1
0, 8, 1, 7 are descendants of 0 7
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6. Expression Trees - * / 3 4 + * 3 2 - 5 6 3 4*(3+2) – (6-3)*5/3
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7. How to construct Expression Trees?
Infix
4*(3+2) – (6-3)*5/3
Shunting Yard Algorithm
Postfix
First Convert the Infix expression into Postfix using Shunting Yard Algorithm.
* 6 3 – 5 * 3 / -
CSC172, Fall 2017

8. How to construct Expression Trees?
* 6 3 – 5 * 3 / - 4 3 2 + * 6 3 - 5 * 3 / -
Assume, each of these values are stored in an array list.
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9. How to construct Expression Trees?4 3 2 + * 6 3 - 5 * 3 / -
Stack
You maintain a separate stack for the construction of Expression Tree. If current element from the array list is a number, create a BTNode (both children as null) and push it to the stack, if the current element element is an operator, create a new Node, pop the last two elements, and make them the right and left child respectively and then push the new BTNode (which is currently holding the tree structure) to the Stack. Continue.
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10. How to construct Expression Trees?* 6 3 - 5 * 3 / - 4 +
Stack 3 2
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11. How to construct Expression Trees?6 3 - 5 * 3 / - *
Stack 4 + 3 2
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12. How to construct Expression Trees?- 5 * 3 / - * 6 3
Stack 4 + 3 2
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13. 5 * 3 / - * -
Stack 3 4 + 6 3 2
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14. How to construct Expression Trees?* 3 / - * - 5
Stack 3 4 + 6 3 2
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15. How to construct Expression Trees?3 / - * *
Stack 4 + - 5 3 6 3 2
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16. How to construct Expression Trees?/ - * * 3 4 + - 5 3 6 3 2
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17. How to construct Expression Trees?- * / * 3 4 + - 5 3 2 3 6
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18. How to construct Expression Trees?- * / * 3 4 + - 5 3 2 3 6
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19. Finally! - * / 3 4 + * 3 2 - 5 6 3 4*(3+2) – (6-3)*5/3
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20. Character Encoding Each character occupies 8 bits
UTF-8 encoding:
Each character occupies 8 bits
For example, ‘A’ = 0x41
A text document with 109 characters is 109 bytes long
But characters were not born equal
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21. English Character Frequencies
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22. Variable-Length Encoding: Idea
Encode letter E with fewer bits, say bE bits
Letter J with many more bits, say bJ bits
We gain space if
where f is the frequency vector
Problem: how to decode?
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23. One Solution: Prefix-Free Codes
c e b a
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24. Why only Binary tree?
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25. Any Tree can be “Encoded” as a Binary Tree
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26. LMC-RS Representation
In this representation, every node has two pointers:
LMC (Left-most-child)
RS (Right Sibling)
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27. LMC-RS Representation
public class Node {
    public int key;
    public Node lmc, rs;
    public Node(int item)
    {
        key = item;
        lmc= rs= null;
    } }
key
lmc rs
CSC172, Fall 2017

28. Tree Walks/Traversals
There are many ways to traverse a binary tree
(reverse) In order
(reverse) Post order
(reverse) Pre order
Level order = breadth first
Tree Walks/Traversals
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29. FULL vs. COMPLETE BINARY TREE
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30. Full Binary Tree 5 2 4 3 Each node is either
Each node is either
(1) an internal node with exactly two non-empty children or
(2) a leaf 8 1
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31. Complete Binary Tree 5 4 2 3 1 9 8 12 7 6 13
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32. Complete Binary Tree
has a restricted shape obtained by starting at the root and filling the tree by levels from left to right.
In the complete binary tree of height d, all levels except possibly level d−1 are completely full.
The bottom level has its nodes filled in from the left side.
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33. Let’s see the examples again
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34. Full vs. Complete Binary Tree5 4 2 3 8 1 5 4 2 3 1 9 8 12 7 6 13
Full but not complete
Complete but not full
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35. Tree using java
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36. A BTNode in Java public class Node { public int key;
    public Node left, right;
    public Node(int item)
    {
        key = item;
        left = right = null;
    } }
key
left
right
CSC172, Fall 2017

37. Inorder Traversal Inorder-Traverse(BTNode root)
Inorder-Traverse(root.left)
Visit(root)
Inorder-Traverse(root.right)
Also called the (left, node, right) order
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38. Inorder Printing in C++
void inorder_print(BTNode root) {
if (root != null) {
inorder_print(root.left);
printNode(root);
inorder_print(root.right); }
“Visit” the node
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39. In Picture 5 3 4 2 4 8 7 3 9 1 5 8 1 9 2 7
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40. Run Time nl = # of nodes on the left sub-tree
Suppose “visit” takes O(1)-time, say c sec
nl = # of nodes on the left sub-tree
nr = # of nodes on the right sub-tree
Note: n - 1 = nl + nr
T(n) = T(nl) + T(nr) + c
Induction: T(n) ≤ cn, i.e. T(n) = O(n)
T(n) ≤ cnl + cnr + c = c(n-1) + c = cn
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41. Reverse Inorder Traversal
RevInorder-Traverse(root.right)
Visit(root)
RevInorrder-Traverse(root.left)
The (right, node, left) order
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42. The other 4 traversal orders
Preorder: (node, left, right)
Reverse preorder: (node, right, left)
Postorder: (left, right, node)
Reverse postorder: (right, left, node)
We’ll talk about level-order later
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43. What is the preorder output for this tree?5 2 4 3 9 8 1 5 4 3 8 7 1 2 9 7
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44. What is the postorder output for this tree?5 2 4 3 9 8 1 3 7 8 1 4 9 2 5 7
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45. Questions to Ponder void inorder_print(BTNode root) {
if (root != NULL) {
inorder_print(root.left);
printNode(root);
inorder_print(root.right); }
Write the above routine without the recursive calls?
Use a stack
Don’t use a stack
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46. Reconstruct the tree from inorder+postorder3 4 8 7 1 5 9 2
Preorder 5 4 3 8 7 1 2 9 5
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47. Questions to Ponder
Can you reconstruct the tree given its postorder and preorder sequences?
How about inorder and reverse postorder?
How about other pairs of orders?
How many trees are there which have the same in/post/pre-order sequence? (suppose keys are distinct)
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48. Number of trees with a given inorder sequence
Catalan numbers
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49. What is a traversal order good for?
Many things
E.g., Evaluate(root) of an expression tree
If root is an operand, return the operand
Else
A = Evaluate(root.left)
B = Evaluate(root.right)
Return A root.key B
root.key is one of the operator
What traversal order is the above?
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50. Level-Order Traversal5 2 4 3 9 8 1 5 4 2 3 9 8 1 7 7
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51. How to do level-order traversal?5 5 2 2 4 4 3 3 9 9 8 8 1 1
A (FIFO) Queue 7 7
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52. Level-Order Print in Java
void levelorder_print(BTNode root) {
// Implement }
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53. Acknowledgement
A lot of the slides are taken from various sources including the Internet, Dr. Hung Ngo’ slides, and various other sources.
CSC172, Fall 2017