The team announced the injuries before the game Saturday against state rival Marquette.
Trice will have surgery on his right foot Monday and won't begin rehab for at least two weeks after the procedure. The sophomore is averaging 9.4 points and playing a team-high 31.5 minutes a game.
King is a promising freshman averaging 5.2 points. He has a left knee injury.
The Badgers are 4-6 and off to a tough start, with four new starters surrounding premier big man Ethan Happ. Their rotation relies heavily on freshmen, and that includes guard Brad Davison, who has been playing through a left shoulder injury.