Yes, if a tree falls on your car you typically have to pay your comprehensive deductible when making your claim for the damages to your vehicle.
Deductibles are not due according to fault; anytime you use a coverage that has a deductible associated with it, you have to pay your deductible. A branch or tree falling on your car would normally be covered under your comprehensive coverage, and the deductible would be due unless your own insurer would waive it for some reason.
If the tree was diseased or had problems that the tree owner should have taken car of, the tree owner may be found negligent and thus pay your deductible amount for your claim. If, however, the tree was healthy and fell due to high winds or other acts of nature, then usually the tree owner won't be held responsible since there was no negligence on their part.