This type of damage may be covered by other insurance, such as your neighbor's homeowner's liability insurance or your own homeowner's policy. If not, you may consider suing your neighbor for the damage to your car cause by his tree's fallen branches.
Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.
