Your neighbors insurance policy should cover the damage to your home, so his first step would be to report the claim to his carrier.
It will handle it and cover the repairs to his home without regard to whether you may have caused the damage.
If his insurance company pays the claim, it will then have the right to collect money from you or from your insurance company. You might have been negligent in failing to remove a tree which you knew was diseased, decayed, dead or in an otherwise dangerous condition.
If you were negligent in failing to remove the tree, then you can be held liable for the damage to his home.
One possible defense you might be able to raise to his insurance company is that the damage was caused by an act of God and not by your negligence.
If you are ultimately determined to have been liable for the damage to his home, your insurance company might cover you for the losses.
For More information afeel free to email me at Bryan@BryanFagan.com or call me at 281-537-2171 or visit my website at www.springdivorce.net