Preprojective algebras of tree-type quivers

Let $Q$ be a tree-type quiver, $\mathbf{k}Q$ its path algebra, and $\lambda$ a nonzero element in the field $\mathbf{k}$. We construct irreducible morphisms in the Auslander-Reiten quiver of the transjective component of the bounded derived category of $\mathbf{k}Q$ that satisfy what we call the $\lambda$-relations. When $\lambda =1$, the relations are known as mesh relations. When $\lambda =-1$, they are known as commutativity relations. Using this technique together with the results given by Baer-Geigle-Lenzing, Crawley-Boevey, Ringel, and others, we show that for any tree-type quiver, several descriptions of its preprojective algebra are equivalent.