Lecture 18 Tree Traversal

1. Lecture 18Tree Traversal CSCI – 1900 Mathematics for Computer Science Spring 2014 Bill Pine

2. Lecture Introduction • Reading • Kolman Section 7.2, 7.3 • Tree traversal • Preorder • Inorder • Postorder • Encoding • Huffman encoding CSCI 1900

3. Tree Traversal • Trees can represent the organization of items • Trees can represent a decision hierarchy • Trees can also represent a process • With each vertex specifying a task • for-loop example for i = 1 thru 3 by 1 for j = 1 thru 5 by 1 array[i,j] = 10*i + j next j next i CSCI 1900

4. i = 1 i = 3 i = 2 j=1j=2j=3j=4j=5 11 12 13 14 15 21 22 23 24 25 31 32 33 34 35 For Loop Positional Tree CSCI 1900

5. Terminology • Traverse a tree • Visit each vertex in a specific order • Visit a vertex • Performing a task at a vertex • Do a computation • Take a decision • Kolman uses the term “search” to mean traverse • Search implies looking for a specific vertex • This is not necessarily the case CSCI 1900

6. Tree Traversal • Applications using trees, traverse the tree in a methodic way • To ensure visiting every vertex, exactly once • We will explore three methodic ways • Example: Assume we want to compute • The average age, maximum age, and minimum age of all of the children from five families • Tree on next slide CSCI 1900

7. Neighborhoodfamilies A.Jones Halls Smith Taylor B. Jones Katyage 3 Tommyage 5 Taylorage 1 Loriage 4 Karenage 14 Mikeage 6 Philage 8 Lexiage 2 Bartage 12 Benage 2 Traversal Example CSCI 1900

8. Traversal Example (cont) • To ensure that we don’t miss a child • Need a systematic way to visit each vertex • To calculate the average, max, and min ages for all of the children • By defining a systematic process • Not only can a human be sure not to miss a vertex • But also can serve as the basis for developing a computer algorithm CSCI 1900

9. One Such Systematic Process • Starting at the root, repeatedly take the leftmost “untraveled” edge until you arrive at a leaf • Include each vertex in the average, max, and min calculations • One at a time, go back up the edges until you reach a vertex that hasn’t had all of its outgoing edges traveled • Traverse the leftmost “untraveled” edge • If you get back to the root and there are no untraveled edges, you are done CSCI 1900

10. Neighborhoodfamilies 1 2 14 7 5 A. Jones Halls Smith Taylor B. Jones 11 3 8 4 10 13 16 Katyage 3 Tommyage 5 Taylorage 1 Loriage 4 Karenage 14 Mikeage 6 6 9 12 15 Philage 8 Lexiage 2 Bartage 12 Benage 2 Vertices Numbered in Order of Visits CSCI 1900

11. Preorder Traversal • The name for this systematic way of traversing a tree is preorder traversal CSCI 1900

12. Preorder Traversal Algorithm • A preorder traversal of a binary tree consists of the following three steps: • Visit the root • Traverse the left subtree if it exists • Traverse the right subtree if it exists • The term “traverse” in steps 2 and 3 implies that we apply all three steps to the subtree beginning with step 1 CSCI 1900

13. Inorder Traversal Algorithm • An inorder traversal of a binary tree has the following three steps: • Traverse the left subtree if it exists • Visit the root • Traverse the right subtree if it exists CSCI 1900

14. Postorder Traversal Algorithm • A postorder traversal of a binary tree has the following three steps: • Traverse the left subtree if it exists • Traverse the right subtree if it exists • Visit the root CSCI 1900

15. A H B C E I K D F G J L Example: Preorder Traversal Resulting string: A B C D E F G H I J K L CSCI 1900

16. A H B C E I K D F G J L Example: Inorder Traversal Resulting string: D C B F E G A I J H K L CSCI 1900

17. A H B C E I K D F G J L Example: Postorder Traversal Resulting string: D C F G E B J I L K H A CSCI 1900

18. Encoding • Encode – translate a string into a series of 1’s and 0’s • ASCII – fixed length code • Code length = 7 bits • Each character is represented by 7 bits • Why 7 and not 8 bits? • Encode 128 distinct characters • 95 printable characters • 33 nonprintable characters CSCI 1900

19. Encoding (cont) • Suppose instead of a fixed length code, we use a variable length • Why? To reduce the size of the encoding • Message compression • Assign the short codes to the more frequently used characters in message • The amount of compression achieved is a function of • The method used to generate the codes • Relative frequency count of the characters in the message CSCI 1900

20. Huffman Encoding • One way of generating the codes • Huffman encoding • Huffman encoding can be represented as a tree • Details of generating the code are beyond the scope of this course • We will examine the use of a generated code CSCI 1900

21. Huffman Tree Example • • • ET • • A I NS 1 0 1 0 1 0 1 0 1 0 CSCI 1900

22. Using Huffman Trees • To convert a code to a string • Begin at the root of the tree • Take the branch indicated by the bit • Look at next bit and take the indicated branch • Until you reach a leaf; this gives the first character • Return to the root and repeat until done • What string is represented by • 1000100 • 11101100101110 • 1110010001 CSCI 1900

23. Using Huffman Trees (cont) • Generate the code for a string • Begin by traversing the tree once, creating a set of <character, character code> pairs • Use the set of pairs to look up the code for each character • What is the encoding for • NEST • STAT CSCI 1900

24. Key Concepts Summary • Tree traversal • Preorder • Inorder • Postorder • Encoding • Huffman encoding • Reading for next time • Kolman Section 7.5 CSCI 1900