Another good question to ask is, “does homeowners insurance cover tree damage to a neighbour's property?” As mentioned, your insurance company will step in if you are found liable for the damage done by the falling tree. Also, the company will defend you if the neighbour decides to sue you. That being said, most policies have coverage limits, and they will only pay up to that amount. After that cap, the homeowner needs to pay out of pocket for damages that go beyond the policy’s limit. So, carefully read through your policy. See if your insurance company covers all claims or only those in which you were not found negligent.
If the damage is minimal, and you want to avoid increasing your premium, consider paying for the damages on your own.