Thanks, everybody. Sorry for the late reply--it's turning out to be a crazy week.
@Michaela G.: That's not my understanding of the law, if the tree was a known hazard. (of course, ianal, but I'm looking at this kind of conservatively.)
@Matthew Paul: I'm tempted to agree with you, but I'm not sure that one necessarily follows the other. A lot of things might not happen over a period of time, but still be a hazard, so this seems like a hard avenue to pursue.
@David Dachtera: The same thing has crossed my mind, as well, and it's a large part of what I'm weighing right now.
@Account Closed: That was one of the first things I asked. :-) PM's response was that the tree seems to have deterioriated in the last two years.
My PM had a couple of tree-removal services come out to take a look at the tree, and one took specific pictures of where he feels the tree is especially sick. So now there's a record of one professional giving his opinion that the tree needs to go. I might be able to find an arborist who could give a contrary view, but I'm not sure where that would leave me in terms of liability, which I'd need to ask a lawyer about. By the time I've factored in the aggravation and expense involved with the lawyer and the arborist, I almost might as well just get the damn tree taken out.