For the neighbor to be found liable, he has to have control over the thing that caused your damage.
The question is, is your neighbor have control over whether the tree would fall?
If you told him the tree is sick or dying otherwise in danger of falling, then he has the control over the tree NOT falling on your house by cutting it down. So if he didn't do anything and tree falls, he's liable.
But if the tree was healthy and it just fell over, it's Act of God. Unless your neighbor is god, you'll have to go through your own insurance.
