demerphq,
Unfortunately, life has side tracked me and I don't have time to respond fully but I did want to leave enough details here so that when/if I get a chance, I will have enough context to respond appropriately.

But that still doesn't make sense to me. Why would you encode all the "HE"'s when "THEN" is also in your dictionary?

In my example, it turns out to be a tie. Compressing the 3 instances of 'HE' as step 1 reduces the string from 12 characters to 9. If instead, as step 1 'THEN' was chosen it would still go from 12 characters down to 9. I should have chosen a non-tie example but the idea was that at that point in time, compressing 3 instances of 'HE' was supposed to be better than 1 instance of 'THEN'

I think to understand how the person arrived at this decision (and why it makes sense to him), you need to follow his logic.

1. Determine the number of distinct 1 byte characters that appear in the file - 96
2. Use only as many bits as necessary to encode the full range of characters - 2^6 = 64 < 96 < 128 = 2^7
3. Use the unused 7 bit sequences to encode common multi-byte sequences - he intended to sample the whole file
4. Compress the file by in a recursive fashion
   1. Choose the sequence that results in maximal compression = $bytes_prior_to_compression - ($num_of_chars_to_encode * $occurrences) + $occurrences
   2. Encode those bytes
   3. Update the frequency data (without re-sampling) and go back to step 1

Also consider that encoding a single THEN wins you more than encoding 3 HE's with the type of scheme you have (4:1 versus 2:1)

Actually, no. Because there is only 1 instance of THEN, encoding it has reduced the file by 3 characters. While it is true that encoding 3 instances of HE also only reduces the file by 3 characters, the example probably should have been 'THE HEN THEN HE HE HE'

Luckily greedy solutions tend to be "close enough" most of the time.

He gave up and went with a heuristic approach that didn't require recalculating the frequency distributions. I didn't see his original approach as viable but I wanted to help him explore it as fully as possible.

If I may be so bold, encouraging people to try to implement algorithms without any domain research is not a bad idea as the starting point of a learning exercise. But I think you should follow up with some proper study of some important algorithms.

That is exactly what this exercise was about. Not everyone agrees with my philosophy of doing things the wrong way to appreciate the right way later.