On October 13, 2019, about 1620 Alaska daylight time, a Cessna 172 airplane, N7267A, sustained substantial damage when it was involved in an accident about 15 nautical miles southeast of the Palmer Municipal Airport (PAAQ) Palmer, Alaska. The private pilot was not injured. The airplane was operated as a Title 14 Code of Federal Regulations Part 91 personal flight. The pilot reported that, after departing the off-airport airstrip near the Knik Glacier, he made a low approach to the same airstrip before continuing the flight to Birchwood Airport (PABV) Birchwood, Alaska. About 15 minutes later, while in cruise flight, the engine began to run rough and the rpm "rolled back." In an attempt to restore engine power, he applied carburetor heat, confirmed a rich mixture, checked the fuel tank levels, and placed the fuel selector from BOTH to the LEFT tank. In addition, he verified the oil pressure, confirmed the magneto switch was selected to BOTH, and that the primer was in and locked. The pilot subsequently selected an all-terrain vehicle trail as a forced landing site. Just before touchdown, about 15 feet above the ground, the left wing struck a small tree and the airplane yawed to the left and touched down in a nose low, off-center angle. Subsequently, the right main landing gear and nose-gear separated, and the right wing struck the ground; the pilot was able to egress the airplane without further incident. The airplane sustained substantial damage to both wings and the fuselage. A post-accident examination of the engine, which included a test run, revealed no evidence of any preimpact mechanical malfunctions or failures that would have precluded normal operation. According to a carburetor icing probability chart, using the nearest weather observation station located about 15 nautical miles away, an airplane operating in the ambient conditions near the accident site would expect carburetor icing at cruise power, and serious icing at glide power.