hi, Mike. The possibility of branches falling onto your house from a dying tree seems like a real concern.
I remember I owned a house once with a tree in the backyard that had a weak branch overhanging the neighbor's yard. I no longer remember the circumstances but somehow there was a requirement after I moved in from the homeowner's insurance inspection that I have the branch removed, because if it had fallen and damaged anything in his yard, I would have been liable, and so I was required to remove it as a condition of obtaining the insurance.
How friendly are you with this neighbor?