The vehicle will be covered if the vehicle has Comprehensive coverage at the time the branch falls on to the vehicle. 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 from the damages the car sustained by the tree branch falling on the vehicle the insurance company will pay up to your vehicle's actual cash value (ACV). This will take your vehicle's depreciation into consideration.
If you do not have Comprehensive coverage on your vehicle than a tree branch falling on it would not be covered. If the tree belonged to someone else you may be able to make a claim under their homeowner's policy, but only if there is negligence found. So if the tree was sick and the owner did not have it cut back or cut down they may be found and fault and their insurance would pay. If however instead the tree was healthy but a wind storm came through and knocked the branch down on your car then typically their insurance would not pay since their insured was not liable for this act of nature.
Remember, you can come here to get a progressive auto insurance quote.