Subtract your loaded vehicle weight from your GVW and thats what you can pull.
...
I would like to correct that to "...thats what you
should not exceed"
The GVWR is the manufacturers guideline as to what the vehicle should be limited to. This factors in a number of engineering, legal, and marketing strategies for selling a truck, but it does not really tell you what the load carrying capacity of the vehicle is.
Naturally, the manufacturer does not wish to be held liable for however much weight a tree service might choose to load onto their truck, so they limit the GVWR a nice safe, sensible load limit.
The legal limitations set by the DOT are actually rather liberal. Most states have a 18,000 pound load limit on a tandem axle, which is just fine for a heavy duty truck. This isn't really true, however, if you try putting 18,000 pounds on your average 1 ton truck. You will break your truck long before you get 18,000 pounds on that light duty axle. Fortunately, the tire load limits protect the axles a little bit from too much weight.
It is also very easy to overload the driveline of a truck, especially since most light trucks have way too much horsepower. If you load your truck too heavy with logs and a chipper, you might find that you quickly shell out your automatic transmission. If you are dealing with a manual transmission, long-term overloading will certainly eat up the gears, u-joints, and differential.