Trees

1. Trees Addenda

2. Huffman Codes • ASCII, EBCDIC (IBM Mainframes) & Unicode use 8 bits for all characters • Morse code, others • variable-length sequences

3. Variable-Length Codes • Each character has: • Has a weight (a probability of ocurrence) • A length • Expected length of a string: • sum of the products of the weights and lengths of all characters in string ABCDE = 0.2 x 2 + 0.1 x 4 + 0.1 x 4 + 0.15 x 3 + 0.45 x 1 = 2.1

4. Decoding • Examine code string • When complete sequence found • Announce recognition of the character • Start decoding next character

5. Immediate Decodability • No code sequence is a prefix of another code (i.e.; every code has a unique start) • Can be decoded without waiting for remaining bits Must decode whole stringD is a prefix of B NO YES

6. Huffman Codes • Immediately decodable • Minimal code length • Need an algorithm • Builds n-bit codes

7. Huffman Encoding • Initialize list of n one-node binary trees T with a weight for each character • Do the following n – 1 times • Find two trees T' and T" in list with minimal weights w' and w" • Replace these two with 1 binary tree whose root is w'+ w" and whose subtrees are T' and T" • label the subtree edges: 0 and 1 • the code for character Ci is the bit string of labels from the root to Ci

8. Huffman Encoding (Example) • Value of each parent=sum of children 1 0 1 .55 0 1 .2 .35 1 0 1 0 .1 .1 .15 .2 .45 B C D A E

9. Huffman Decoding • Initialize pointer p to root of Huffman tree • While not end of message string: • a. Let xbe next bit in string • b. if x = 0 set p = left child pointer else set p = right child pointer • c. If p points to leaf • Display character with that leaf • Reset p to root of Huffman tree e.g.; code string: 0001011010 B E A D