How does the death penalty apply to Romeo and Juliet?

In Act I, Scene 1 of Romeo and Juliet there is a major brawl between the Montagues and Capulets. It begins with the Capulet servants "biting their thumbs" (a severe insult) at their Montague counterparts. It escalates as Tyblalt, despite the attempts by Benovlio to mitigate the situation, joins in and says he hates the Montagues as much as hell. The scene provides exposition for the remainder of the play.

The battle in the street...

In Act I, Scene 1 of Romeo and Juliet there is a major brawl between the Montagues and Capulets. It begins with the Capulet servants "biting their thumbs" (a severe insult) at their Montague counterparts. It escalates as Tyblalt, despite the attempts by Benovlio to mitigate the situation, joins in and says he hates the Montagues as much as hell. The scene provides exposition for the remainder of the play.

The battle in the street is eventually broken up by the Prince, who rules Verona. He is obviously fatigued by the continual outbreaks of violence in the city streets, so he issues an edict which outlaws the fighting between the two families and threatens the death penalty for anyone who breaks his proclamation. He says,

Three civil brawls bred of an airy word
By thee, old Capulet, and Montague,
Have thrice disturbed the quiet of our streets
And made Verona’s ancient citizens
Cast by their grave-beseeming ornaments
To wield old partisans in hands as old,
Cankered with peace, to part your cankered hate.
If ever you disturb our streets again,
Your lives shall pay the forfeit of the peace.

Later, however, the Prince reconsiders his proclamation when Tybalt kills Mercutio and Romeo, in a fit of revenge, kills Tybalt. After hearing the story of the fight, which breaks out in Act III, Scene 1, from Benvolio he sentences Romeo to banishment rather than death. The Prince listens to the pleas of Lord Montague and agrees that exile is the just decision. He says at the end of the scene,

And for that offense
Immediately we do exile him hence.
I have an interest in your hearts’ proceeding:
My blood for your rude brawls doth lie a-bleeding.
But I’ll amerce you with so strong a fine
That you shall all repent the loss of mine.

His decision may have also been influenced by the fact he was related to Mercutio, as he is to Count Paris, who dies later at the hands of Romeo outside the Capulet tomb.