Two characterizations of Bass orders via branches

It has been known for some time that the orders in the four dimensional algebra of two by two matrices over a local field that can be written as a finite intersection of maximal orders are precisely those whose Gorenstein closure is Eichler. In this paper, a similar characterization is given for orders whose Gorenstein closure is a Bass order. A second characterization, this time for the Bass orders themselves, is given in terms of their branches, i.e., maximal subgraphs of the Bruhat-Tits tree whose vertices are orders containing them.