If the tree fell due to weather, the neighbor is not liable so it would have to be filed under your own homeowner's insurance. The neighbor is only liable if you can prove the tree was dead or dying and they did nothing to remove the problem.
You should discuss this issue with your homeowner's insurance
company. Generally, you are responsible for the portion that lies
on your property unless you can prove your neighbor's negligence
caused the tree to fall. Acts of nature are beyond our control. If
a tree fell on your house whether from your yard or some other yard
due to a natural occurrence you should be covered under you own
homeowners policy for the damage.
A tree owner must act reasonably to prevent trees from damaging the neighbor's property. Ignoring a damaged or unhealthy tree could create liability. If the tree was obviously dead and you had prior notice from obvious observation or actual notice from the neighbor you would owe a duty to abate that danger or hazard. Your failure to abate the danger may leave you legally responsible for the damage.
If you had notified your neighbor in writing of the possibility of damage from the tree and requested to have the tree trimmed or removed due to a dangerous condition, and your notice was ignored, then the neighbor may be liable for damages. A tree owner must act reasonably to prevent trees from damaging the neighbor's property.
If your neighbor's tree is in a dangerous condition, you should send a letter by certified mail to the neighbor (and if possible to their insurance company and yours) stating that they need to cut the branches or remove the tree to avoid any potential damage to your property. You should take some photos and keep them with a copy of the letter and the green card as proof of delivery.
Felled and Damaged Trees
The responsible party would be Mother Nature. Unfortunately she does not carry liability insurance.
In general, no one is responsible or financially liable for acts of nature as they are beyond our control.
If your neighbor's tree or your own tree falls on your house, your insurance will cover you. If your tree gets blown into your neighbor's house, the same applies. His home insurance would cover him. It does not matter who's tree it was.
If the damage is that the tree is down somewhere out on the property, then it just depends on whether you have debris removal or landscape restoration on your policy. Otherwise each homeowner is responsible for the removal of that portion of the tree that fell on their own property.