This was the pilot's first time at this airport. He was invited to land by the airport owner who radioed and talked him through the landing. He landed successfully, and the owner then took him up for a flight to show him the normal takeoff and landing for this airport. After the second landing, the owner got into his airplane and asked the pilot to follow him. During the takeoff, the pilot stated that the engine seemed to be losing power and not climbing as normal. A witness who saw the airplane takeoff reported that the engine did not seem to be developing full power. He added that the airplane only seemed to gain 50 to 75 feet when the pilot made a right turn and disappeared behind the trees. The airplane collided with trees on top of the hill. According to the pilot operating handbook, it is recommended that for takeoff the carburetor heat should be off. Examination of the airplane by the FAA revealed the carburetor heat was in the on position.