How do I get my neighbor to cut down his trees that are hanging over my property?

You may cut down any overhanging limbs that are hanging over your property line

Start by asking his insurance company to pay for the damage to your roof. I suspect that is quicker and more effective than anything that you can do through court. You cannot get him to cut down the trees; you can get him to cut off the overhanging limbs.

Your insurance company might pay your damages and then pursue your neighbor's insurance company. If not, you have a potential trespass and nuisance claim against your neighbor. Have you received any estimates on the cost of repairing the damage?