THE FLT REPORTEDLY INVOLVED MOUNTAIN FLYING TECHNIQUES. THE STUDENT OBSERVED THAT THE ACFT WAS NOT CLIMBING NORMALLY & THE CFI SUGGESTED THEY SLOW TO 70K. THE STUDENT STARTED A TURN TOWARD A WIDE VALLEY BUT THE CFI URGED HIM TO CONTINUE TOWARD THE MTNS WHICH TOOK THE ACFT INTO A NARROWER VALLEY & RISING TERRAIN. AFTER THE STUDENT REPORTED THAT THE ACFT WAS NOT CLIMBING AT ALL THE CFI TOOK CONTROL. HE WAS UNABLE TO RE-ESTABLISH A CLIMB & REPORTED THAT A TURN-AROUND WAS NOLONGER A SAFE MANEUVER. HE SLOWED THE ACFT & MUSHED INTO THE TREES. THE CFI REPORTED HE ADDED CARB HEAT WHEN THEY WERE NO LONGER ABLE TO CLIMB, BUT AFTER NOTICING AN RPM DROP IN 15 SECS HE SWITCHED CARB HEAT TO COLD. HE SLOWED THE ACFT TO 57-58K & EXTENDED 10 DEG OF FLAPS. THE C-172P POH INDICATES THAT THE BEST ANGLE OF CLIMB SPEED(VX) VARIES FROM 60-65K FROM SL TO 10,000 FT. IT SUGGESTS FLAPS UP TO OBTAIN BEST ANGLE PERFORMANCE. THE ACFT CAME TO REST INVERTED WHICH LEFT THE OCCUPANTS INVERTED & THEY REPORTEDLY HAD DIFFICULTY IN RELEASING THEIR SEAT BELT BUCKLES.