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
Char
Code
Length A 01 2 B
1000 4 C
1010 D
100 3 E 1
Character A B C D E
Weight .2 .1
.15
.45
ABCDE = 0.2 x x x x 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 string D is a prefix of B NO
YES
Char
Code
Length A 01 2 B
1000 4 C
1010 D
100 3 E 1
Char
Code
Length A 01 2 B
1000 4 C
0001 D
001 3 E 1

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 1
.55 1 .2
.35 1 1 .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 x be 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:
B E A D