If the damages caused to the house which is insured, by falling trees, are included in the terms of insurance; then the homeowner’s policy pays for the damages. It depends on the perils mentioned in the coverage. It is the liability of the Insurance Company to pay you for the damages.
It does not matter to you whether the neighbor’s policy covers it or not. In fact, even if your neighbor does not have any home insurance policy, it does not jeopardize your claim. All you need to do is inform your insurance company about the damages. Their assessor will assess the damage and take appropriate action thereafter.
In case tree damage is not covered then it is stated in the policy terms and conditions that it don’t cover properties which has trees in their near vicinity. In such cases, these exclusion needs to be added with additional premium. For premium, the height and number of the trees, and the distances from the property are taken into consideration.