The pilot stated he was on a training flight when, during the third landing, he felt the airplane pull to the right and start vibrating upon touchdown. He attempted to correct to the left, but the airplane continued to the right, exited the runway, and then turned back to the left before stopping. The right aileron was substantially damaged during the event. The right tire was found deflated. Examination of the deflated tire found it was worn with little remaining tread. The exact location of a leak could not be determined due to damage to the tire. The tire was installed about 20 months and 375 flight hours prior to the accident. The number of landings on the tire was not determined. The pilot stated he thought there was sufficient tread remaining on the tire prior to the flight and he intended to replace the tire during the next annual inspection. It is likely the worn tire deflated prior to the accident landing due to excessive wear.