How do I hold treecutter responsible for damage done to my property?

You can sue the company and the individual, who damaged your driveway for negligence.  Both the company and individual would be named as defendants in the lawsuit.

Prior to filing the lawsuit, try to obtain the name of the insurance carrier.  If you obtain the name of the insurance carrier, file a claim with the insurance carrier based on the cost of repairs to your driveway.  If  the claim can be resolved with the insurance carrier, you won't need to file a lawsuit.  If the company/ individual, who damaged your driveway, refuses to provide the name, address, and telephone number  of the insurance carrier, proceed with the lawsuit.