No, neither your liability insurance nor the liability insurance of your friend would cover damages to your vehicle from a one car accident where it hit a tree. For your automobile to be covered for the damage from this incident you would need physical damage collision coverage.

Property Damage Liability does not cover the vehicle which you are driving but instead covers you if your car damages someone else's property. Usually it is their car, but it could be a fence, a house or any other property damaged in an accident. So if your friend had damaged another person's fence or car when he crashed your car then your liability insurance would have covered that other person's damages. And if your liability coverage limits were exceeded then his liability insurance would have been used as excess or secondary coverage.

If you have collision coverage on your vehicle then you should be able to put a claim in, even though your friend was driving at the time of the accident. Collision covers damage to your car when your car hits, or is hit by, another vehicle, or other object. Pays to fix your vehicle less the deductible you choose.