Question: If a tree in a neighbor’s yard falls and damages a fence, a car, your roof…is it their responsibility to cover the damage?
Answer: Long story, short. No. It is not their fault that the tree fell. In most cases the tree was there before the fence was put up. It is understandable that trees get larger over time and the fence may have not been close. However, trees tend of fall over when the wind blows of it is heavy with ice and can fall into your yard.
If the tree is healthy and there would be no danger of it falling other than outside sources(i.e. weather or other circumstances out of the norm) in most cases there would be no negligence on the part of the neighbor if the tree does fall.
While the above is the most common scenario. If you are in a situation where you note that there is a diseased, dying or dead tree that poses a danger and the owner is aware, there could be liability. However, you would mostly likely need prior documentation. Send a certified letter that there is a diseased or dead tree that could potentially fall and damage your property or injure someone. Send a certified letter and keep a copy and the mail receipt. You can also document with photos. Liability is often denied if the tree is near the back of the lot and the owner says they were unaware. That is why the above documentation would be needed and would be the difference between you covering the damages or your neighbor(or his insurance company)taking care of the damages.
Most neighbors want to be, well, neighborly, and would address any potential dangers or hazard in conversation. But you may want o follow up a with a friendly written request to remove the tree.