The student pilot reported that while on the downwind leg of the traffic pattern, after crossing abeam the runway touchdown zone, he inadvertently shut off the fuel valve instead of activating the carburetor heat. Unaware of his mistake, the pilot continued on the landing approach believing the engine was operating at a reduced power setting. The pilot reported that after maneuvering the airplane onto final approach, he realized that the airplane was lower than normal and attempted to increase engine power. After the engine did not respond to his throttle movements, the pilot noticed that the carburetor heat control was not turned on. Believing the engine was experiencing carburetor icing, the pilot activated the carburetor heat. Still without any response from the engine, the pilot completed a forced landing short of the intended runway. The airplane collided with two trees during the forced landing, which resulted in substantial damage to both wings. While securing the airplane after the accident, the pilot realized that he had inadvertently shut off the fuel valve while on the downwind leg instead of activating the carburetor heat. He stated that the two systems had similar push/pull control knobs that were positioned next to each other in the cockpit.