Presume = to accept as true in the absence of proof to the contrary (ie, in the legal world, accused people are presumed innocent until proved otherwise).
Presume = to suppose to be the case on the basis of probability (I couldn't see her in the office, so I presumed she had left for the day).
Assume = to suppose to be the case without any proof/evidence at all (I assume that Martians are tall and grey).
It is easy to get them mixed up and difficult to use them correctly.