Postgresql full text search part of words

Sounds like you simply want wildcard matching.

One option, as previously mentioned is trigrams. My (very) limited experience with it was that it was too slow on massive tables for my liking (some cases slower than a LIKE). As I said, my experience with trigrams is limited, so I might have just been using it wrong.
A second option you could use is the wildspeed module: http://www.sai.msu.su/~megera/wiki/wildspeed (you'll have to build & install this tho).

The 2nd option will work for suffix/middle matching as well. Which may or may not be more than you're looking for.

There are a couple of caveats (like size of the index), so read through that page thoroughly.

You can use tsearch prefix matching, see http://www.postgresql.org/docs/9.0/interactive/textsearch-controls.html#TEXTSEARCH-PARSING-QUERIES

postgres=# select to_tsvector('tree') @@ to_tsquery('tr:*');
 ?column? 
----------
 t
(1 row)

It will only work for prefix search though, not if you want partial match at any position in the word.

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