Richards, 62, suffered a mild concussion while on vacation in Fiji Thursday. Several Australian and New Zealand media outlets reported the fall from the tree.
New Zealand's Sunday Star-Times also reported that Richards got on a Jet Ski after the fall and had another accident.
Richards was treated in a Fiji hospital before being airlifted to Auckland's Ascot Hospital for observation. It was not immediately clear exactly when he was released or where he went afterward.
"I can categorically confirm Mr. Richards ... is no longer a patient in this hospital," hospital duty manager Geoff Sparks told The Associated Press on Tuesday. "He has been discharged.
Band spokeswoman Fran Curtis has said only that Richards was injured earlier this week and flown to New Zealand with his wife, Patti, for observation. Her statement Saturday did not comment on Richards' condition or how he was injured.
Richards and his wife were staying at the exclusive Wakaya Club resort on a small Fijian island when he was injured.