Hire a lawyer to sue the homeowner neighbor. He/she can deal with their insurance company, it's not your problem. One of them has to pay. Don't waste your time anymore. Use the "find a lawyer" tool on AVVO.
We do not have an attorney-client relationship. I am not your lawyer. The statements I have made do not constitute legal advice. Any statements I have made are based upon the very limited facts you have presented, and under the premise that you will consult with a local attorney. This is not an attempt to solicit business. This disclaimer is in addition to any disclaimers that this website has made. I am only licensed in California.
The owner of the tree is responsible for damage caused by the tree. I agree that you should contact your insurance company, his insurance company, send him a written demand by certified letter.
You may also want to contact a lawyer, but there is no reason to delay submitting insurance claims.