The flight instructor reported that, while the student pilot was conducting his fourth takeoff during his first flight lesson, the airplane encountered a wind gust and entered a “sharp” left turn toward a tree line. The flight instructor took control of the airplane, verified the throttle was at full power, and adjusted the airplane’s pitch for best angle of climb airspeed. He thought the airplane had cleared the trees; however, the aft fuselage or empennage struck some branches causing the airplane to lose airspeed. The airplane entered a descent and impacted a field on the other side of the tree line. The airplane sustained substantial damage to the left wing, left horizontal stabilizer, and the fuselage. The flight instructor reported that there were no preaccident mechanical failures or malfunctions with the airplane that would have precluded normal operation. At the time of the accident, the student pilot was taking off on runway 30 with wind from 270o at 11 knots with gusts to 15 knots.