[Lord Syslox of Emyn Beverli]

Dont know if im right, ive got no evidence, just ideas:
The Mayor of the Shire (or more specifically of Michel Delving) presided over important events as well as smoothing out any problems between inhabitants (i imagine). The Master of Buckland would have the same role but just for Buckland (it being almost independant). The Thain was chosen to represent the King (after the end of the North kingdom) so he must have had some governing power. Consider the Sherifs and the posthobbits, someones gotta organise (and originally ordain) them. In the appendixes it says King Elessar made the Thain, the Master and the Mayor councellors of the North Kingdom. This means they (fundamentally) represented the Shire people in the King's Court. When the Tower Hills were given to the Hobbits, the Fairbaines became the Wardens of the Westmarch. The Mayor would have taken care of the westfarthing (thats were Michel Delving is), the Master of Buchland had the eastfarthing (obviously), and the Thain of Tukborough (in the southfarthing) would have shared the north and south with the other two.

To conclude:
I imagine Bree would have a system similar to the shire hobbits, were a number of individuals dont rule but act almost as judges and help each other out.