If you are still paying for your vehicle, then your lien holder may require you carry comprehensive and collision coverage.Otherwise the need to carry physical damage coverages or comprehensive and collision is a personal choice depending on your financial situation and the value of your car.
State laws and insurance policy terms differ however in general in most states if someone else hits your vehicle and is found liable for the damages done to your car then normally their property damage liability coverages would cover the cost of the repairs to your vehicle. However some states do not require property damage liability and other states have their own unique insurance laws such as Michigan where there is the mini-tort so that the most you can collect from an at fault party is $500.
You many also find that you would want collision coverages in case your car is hit and the driver does not stop or leave a note (a hit and run) or an accident where your car is damaged by an uninsured driver and thus would need to use your own collision coverage. You could also be in a position where you are in an accident and the other party's insurance does not find their policyholder at fault and thus you would need collision coverage to cover your own vehicle for the damages sustained.
Comprehensive or "other than collision" would be need for fire, theft or as you mentioned if a tree falls on your vehicle. When placing a claim with your collision or comprehensive coverages there would be a deductible to pay.
If you have a newer car then normally you would carry collision and comprehensive on the vehicle while you were paying off the lien holder and then it would be up to you to decide if you wanted to continue these coverages or not. Many people decide to do so until the cost of carrying these physical damage coverages cost more than or equal to what the car would be worth if it was found to be a total loss.
Get a car insurance quote with us for your state's minimum coverages or the minimum coverages plus collision and comprehensive coverages so that your vehicle is better protected.