General aviation pilot reported a near miss with another aircraft when entering the traffic pattern at a non-towered airport. The pilot maneuvered to avoid the aircraft.

Narrative

While entering downwind for Runway 12 at TRM; at 1000'; I received an alert from FOREFLIGHT; of extremely close traffic.Foreflight showed faster traffic coming from behind me; climbing; 300' below.I saw the aircraft; approaching me from behind; 30* angle; 300' and climbing into me; while I was in the pattern. The aircraft was at a fast rate of speed; flying thru TRM's traffic pattern.I could look down to my left; and SEE the pilot in the cockpit; but he never changed path.I immediately climbed; banking left to avoid the impending collision.I had made advisory position calls at 10 miles; 5 miles; and 3 miles; of my position and intended landing runway.I had my Navigation lights;Tail Beacon; Landing and Taxi Lights; and my wingtip Strobe lights all on.I attempted to contact the other pilot on CTAF; with NO response.The other plane was at approximately 700' in altitude at the time of the near miss; and overtook my 110 Kt airspeed; from behind; flying thru the pattern for both runways at TRM.The other plane did not land at TRM.Had Foreflight not advised me of the dangerously close; fast traffic; there WOULD have been a collision!The lack of a tower at this airport; the failure to monitor the CTAF; flying thru the traffic pattern; at a fast rate of speed; and failure to look out the window; all contributed to this near miss.