Can a dean expel a doctoral student without a trial?
Universities do not in my experience hold "trials" in order to reach their decisions, however weighty.  So the answer to the literal question asked is probably "yes".  

I guess what you mean to ask is whether the dean has the unilateral power to do this.  I'm not entirely clear on what "this" is: what does "preventing him from completing his PhD" or "effectively expelling" mean, precisely?  But even if I did, I would have to know the rules of your friend's university rather intimately in order to answer.  (Someone in your university has the power to do this.  As @Paul comments, probably more than one person was involved in the decision.  Just because the action looks single-handed to your friend does not mean that other university officials were not involved.)

One tip: if your friend's adviser doesn't know, get your friend to tell her!! (i) Could it make things any worse?  (ii) Won't she find or sooner or later?  Sooner may be soon enough to at least try to do something about it; later, maybe not.

Added "It seems the dean is harassing him, pure and simple." Well then he should report it to....oh.  Seriously, if by this you mean that you think the dean has some kind of vendetta against your friend which caused him to simply fabricate these charges: though obviously I don't and can't know the situation, I find that very unlikely.  Though there may be no "trial" system in the university, there will be some kind of clear guidelines and procedures for expelling students.  If the harassment is simply made up then the expelling couldn't possibly have followed these procedures, which would open the university up to a trial, possibly an embarrassing and costly one.  I think I understand this clearly, but a dean understands it like I can't even imagine.