Shortly after takeoff, the engine lost partial power. The CFI stated he applied carburetor heat, and the engine lost more power. He stated he turned the carburetor heat off and the engine regained some power and then continued to lose power. He stated he switched magnetos, but nothing helped to restore power. He executed an emergency landing in a field below his flight path. The airplane slid along the snow covered ground until it impacted a row of trees. The airplane and engine were examined with no anomalies found. At the time of the accident, Dayton weather facility was reporting a temperature of 19 degrees F and a dew point of 11 degrees F. The pilot stated that prior to the flight, the airplane was covered with snow and ice. He stated he towed the airplane into a hangar and heated the airplane for 45 minutes. He stated he heated the airplane until all the ice and snow was melted and then moved the airplane outside. He said the airplane was outside about 10 minutes before they started the engine.