The indecomposable preprojective and preinjective representations of the quiver ~D_n

Consider the quiver ~D_n and its finite dimensional representations over the field k. We know due to Ringel in that indecomposable representations without self extensions (called exceptional representations) can be exhibited using matrices involving as coefficients only 0 and 1, such that the number of nonzero coefficients is precisely d-1, where d is the global dimension of the representation. This means that the corresponding ''coefficient quiver'' is a tree, so we will call such a presentation a ''tree presentation''. \nIn this paper we describe explicit tree presentations for the indecomposable preprojective and preinjective representations of the quiver ~D_n. In this way we generalize results obtained by Mr\\' oz for the quiver ~D_4 and by Lorinczi and Szanto in for the quiver ~D_5.