Derived-tame blowing-up of tree algebras

Let k be an algebraically closed field and A be a tree algebra. We consider here a class obtained by the blowing-up of a tree algebra A at a set of vertices D of A, such an algebra is denoted by A{D}. The objective of this paper is to prove the equivalence between the derived-tameness and the non-negativity of the Euler form for algebras of this form. We also show that, in this case, if D is a non-empty set then A{D} must be derived equivalent to a special incidence algebra, called semichain algebra.