The pilot reported that the engine gradually began to lose power just after takeoff. The airplane began to descend, and subsequently crashed into trees near the end of the runway. A post-crash exam of the aircraft revealed that the carb heat control was in the 'hot' position. The pilot stated that he did not use carb heat during the flight, or during pre-flight run-up. Further exam of the aircraft after removal from the crash site revealed impact-related marks on the carb-heat control push-pull rod system, which indicated that the control was in the 'hot' position at the time of impact. The carb heat control was not readily visible to the pilot, due to its location. Examination of the engine revealed no evidence of any pre-existing mechanical malfunction or failure. No evidence of contamination was found in the airplane fuel supply or system. The ambient air temperature was reported to be 80 degrees; the density altitude was estimated to be 2700 feet.