This paper is a continuation of [6], where irreducibility in the sense of Duffus and Rival (DR-irreducibility) of monounary algebras was defined. The definition is analogous to that introduced by Duffus and Rival [1] for the case of posets. In [6] we found all connected monounary algebras A possessing a cycle and such that A is
DR-irreducible. The main result of the present paper is Thm. 4.1 which describes all connected monounary algebras A without a cycle and such that A is DR-irreducible.Other types of irreducibility of monounary algebras defined by means of the notion of a retract were studied in [2]–[5].