Normally a landowner does not have responsibility for people outside their land, if the danger is not from his own activity. HOWEVER, all jurisdictions in this country apply an BIG exception, the urban tree doctrine, that the landowner is liable if the injury or damage is proximately caused by his negligent maintenance of his trees.
One more thing, insurance is treated as a collateral source at law. It means, that your neighbor will not only get insurance money, but also get recovery from you by law, the same as in car insurance.
One more thing, insurance is treated as a collateral source at law. It means, that your neighbor will not only get insurance money, but also get recovery from you by law, the same as in car insurance.