At the end of the first scene of Act 3, the Prince banishes Romeo saying that if he tries to return, as soon as he's found, he will be put to death.  The Prince does not sentence Romeo to death for the killing of Tybalt because Romeo's mother and father plead with the Prince.  They remind the Prince that Tybalt started the skirmish and that Tybalt killed Mercutio, Romeo's friend.  Romeo was simply avenging Mercutio's death. They also point out that Romeo is not known to be a brawler; in fact, Romeo tried to stop the fight between Tybalt and Mercutio.