Applications of Trees

ICS 211 Spring 2000

Applications of trees

Represent organization
Represent computer file systems
Networks to find best path in the Internet
Chemical formulas representation
Outlines, etc…

More?

Ordered storage to be used in binary search
Decision trees
Encoding

Encoding the alphabet using binary digits.

26 letters

2^4 = 16 (not enough)

2^5 = 32 (some to spare)

A = 00000
B = 00001
C = 00010
D = 00011……

Here, you know that if every letter is represented by 5 digits so you will know exactly what has been encoded. Great! problem solved… BUT this has some disadvantages… Some letters repeat themselves more times than others, if those high-frequency letter’s representation were shorter, we could save:

Storage resources
Transmission time
Maybe even money…..$$$$

Lets now take into consideration the frequency of the letters.

The vowels are more frequent than the letter "z" and yet we use the same amount of digits to encode either one of them…

One possibility is to use variable length code.

Example:

E = 0;
A = 1;
T = 01;

Since T is less frequent than E or T.

PROBLEM

If we have 0101 it can translate to either:

EAT
TEA
EAEA, or
TT

Too bad L , so sad…

But, there are Prefix Codes ! J

What the $#%$ is a Prefix code?

Prefix Code: A code that has the property that the code of a character is never a prefix of the code of another character.

Here is a Prefix Code example:

E = 0;
A = 10;
T = 11;

10110 = ATE and it can’t correspond to anything else.

Why? Because 1 is a prefix and by itself it doesn’t represent a character, while 10 does.

Examples:

By sight, is this a prefix code? Please ignore that all letters are encode using the same length

A = 11
E = 00
T = 10
S = 01

Write: EATS, SEAT, TEA

What about… is this a prefix code?

A = 0;
E = 1;
T = 10;

Write: EATS, SEAT, TEA

What about… is this a prefix code?

A = 101;
E = 11;
T = 001;
S = 0001;

Try it with: EATS, SEAT, TEA

How do we go from prefix codes to binary trees?

A prefix code can be represented using a tree, where:

This tree will be a binary tree.
Left side edge will be labeled 1
Right side edge will be labeled 0
Leaves contain characters
Internal nodes contain empty strings
Characters are encoded with the bit string constructed using the labels of edges in the unique path from the Root to the leaves

Example:

A = 11;
E = 0;
T = 101;
S = 100;

Remember: Left is 1, right is 0. However this is just a convention to follow your book. Other books show the opposite. Check p.301 Standish

Ok.

We had the tree and we use it to decode. Fine!

Now…

If we had letters and a given frequency for each one of the letters, how do we build a tree?

Your book pages 298-300 There is an explanation for the Huffman code and an example. Try following your book’s example and build the real Huffman Code prefix binary tree using the real letter frequencies .