This information is given for educational purposes only. No attorney-client relationship exists between us.
Your neighbor is not liable for the damage to your property UNLESS he was negligent in the maintenance of the tree. If he failed to trim dead branches or allowed it to lean in your direction, then he is negligent and liable for the damages. You can ask for his insurance contact info and make a claim. If he won't give you the info, you can file a small claims suit up to $10,000. If there was no negligence, then he is not liable to you and should file a claim on your homeowners insurance for the damage.