The tame dimension vectors for trees

We consider representations of quivers over an algebraically closed field K. A dimension vector of a quiver is called hypercritical, if there is an m-parameter family of indecomposable representations for the dimension vector with m?2, but every family of representations for all smaller dimension vectors depends on a single parameter. We characterise the hypercritical dimension vectors for trees via their Tits forms and those of their decompositions and present the complete list of the hypercritical dimension vectors.Finally, this leads to a combinatorial classification of the tame dimension vectors for trees which is also given by the Tits forms.