I cut down a big tree and it fell on my truck. Will insurance cover that?

Your insurance should cover the damages minus your deductible. In this case, you would file a comprehensive claim for the damage. A comprehensive claim does not raise your rate as much as a collision would — about 4.35% for your first comp claim.

As a warning, they may investigate and take extra time for the claim to make sure there was no foul play. I would keep photos of the damage and try your best to document everything that happened. 

If you have any questions, don't hesitate to ask.