So I would say to that - if from the outside this tree appeared diseased and dying, AND, that was what caused the tree to fall (I have seen healthy portions of a tree fall during a savage windstorm, and had nothing to do with what was going on in other portions of the tree…) then when you submit that claim to your insurance company, your insurance company may have surrogates (or tried to) against your neighbor’s insurance company.

I am a lawyer who has dealt with large trees knocking over neighbor’a houses, taking out cars, and in one instance crushing a man as he was driving by in a bad storm of all things.

This is the correct answer. If the limb overhangs your property then you can cut it down no matter it’s condition. If the limb is diseased or dying, the person who owns the land where the base of the tree sits is aware of this (or has reason to be aware) and that limb falls because of the diseased condition - then the neighbor could be liable.

Either way. To anyone who has a limb overhanging your car or property - if you want it gone then cut it down. Best to talk to your neighbor about it first to explain this before doing it.