While phrasal verbs are often, if not mostly, unpredictable regarding the semantic function of the particle (up, down, on, etc.), there are a few patterns. Namely, in the instance Martha mentioned–”cut up a tree“–the particle up often connotes completeness.
Compare:
tear a sheet of paper
vs.
tear up a sheet of paper.
Other phrasal verbs are used to make this distinction: cut up, chop up, clean up, wash up, etc.
Anyone know of others that generally follow patterns?