The student pilot reported that, during landing on a private grass airstrip, the airplane bounced so he added power to stabilize the airplane before continuing the landing. However, they "began to run out of runway," so he added full power and the flight instructor took control of the airplane. The airplane struck the top of some trees and the right wing hit another tree. The instructor reported that he believed the airplane's throttle was improperly rigged, explaining that if you released the throttle, it would spring to full open. The instructor added that during landing the airplane bounced, the student "went wide open throttle," and the airplane stabilized about 15 feet above the ground. He assumed that the student was performing a go around, although they had not previously discussed the procedure. The airplane continued toward the end of the airstrip, but the student did not perform a go around or land. The instructor took control of the airplane and initiated a go-around, but the airplane impacted trees beyond the end of the runway. The airplane sustained substantial damage to the fuselage and both wings. The flight instructor reported that there were no preaccident mechanical failures or malfunctions with the airplane that would have precluded normal operation. The instructor recommended more practice at a conventional airport prior to practice at the private airstrip with restricted maneuvering space.