I am not a lawyer and I can't comment on weather the original poster has to pay the bill or not.

However, I believe you are within your rights in Massachusetts to trim anything that is on your property - weather it originates from there or not. We were told we could basically chop our neighbors tree off at the property line if we wanted to. The neighborly thing to do is ask, but I don't think the OP is in any danger from having trimmed branches of a neighbors tree that were touching his roof.