THE ACFT COLLIDED WITH TREES DURING TAKEOFF. THE CFI AND STUDENT HAD MADE A LANDING ON A RWY THAT WAS 2/3 COVERED WITH SNOW. THEY THOUGHT THEY COULD MAKE A SHORT FIELD LANDING AND A SHORT FIELD RUNNING TAKEOFF. THEY NOSED DOWN IN THE SNOW DURING THE ATTEMPTED TAKEOFF. ASSESING THE DAMAGE IN THIS MISHAP AS MINOR THEY RESTARTED THE ACFT AFTER PUSHING IT OUT OF THE SNOW. A TAKEOFF FROM THE CLEARED 1/3 OF THE RWY TOWARD THE SNOW RESULTED IN AN ABORT IN THE SNOW. THEY DECIDED TOTAKEOFF FROM THE SNOW COVERED AREA TOWARD THE CLEARED END OF THE RWY. THIS LEFT NO ROOM FOR AN ABORT. WHEN AIRBORNE, USING FLAPS BUT OUT OF GROUND EFFECT, THE ACFT SETTLED INTO THE TREES. THE ARPT IS CLOSED BY SNOW IN WINTER MONTHS WITH NO SNOW REMOVAL PLANS AND IS SO NOTED IN THE GUIDE TO ARPTS.THE AREA CLEARED BY THE WIND AND USED BY THIS FLT WAS ABOUT 750 FT LONG.