Normally the damage to your car from the tree falling on it should be covered by your auto insurance. The vehicle will be covered if the vehicle had comprehensive coverage on it when the tree caused the damage. Comprehensive covers your vehicle for losses resulting from incidents other than collision. It pays to fix your vehicle less the deductible you choose. If your vehicle is deemed a 'total loss,' the insurance company will pay up to your vehicle's actual cash value.
Homeowner's insurance pays for damage done to the residence or any other property coverage is extended to such as an unattached structures or your personal property and provides liability coverage (if, for example, you tree hurt someone else or damaged their property). Damage done to automobiles is typically specifically excluded under a homeowners policy.
If instead it was your neighbor's tree that fell on your car instead of a tree from your own property coverage may or may not exist under their policy; one factor there will be whether or not their tree was alive at the time of the loss. For example, if your neighbors knew the tree was diseased and did not remove it within a reasonable time period, they should be found liable and their policy should cover the damage. Same would hold true for you if your tree fell on a neighbor's car.
If you do not have comprehensive coverage on your vehicle then you can read through your homeowner's policy and speak to your insurance agent to see if the car's damage would be covered or not since the terms of policies can differ depending upon state laws and insurance companies' guidelines.