  We deduce a necessary condition for Auslander-Reiten components of the
bounded derived category of a finite dimensional algebra to have
  Euclidean tree class by classifying certain types of irreducible maps in the
category of complexes. This result shows that there are only finitely many
Auslander-Reiten components with Euclidean tree class up to shift. Also the
Auslander-Reiten quiver of certain classes of Nakayama are computed directly
and it is shown that they are piecewise hereditary. Finally we state a
condition for $\Z[A_{\infty}]$-components to appear in the Auslander-Reiten
quiver generalizing a result in \cite{W}.
