  We introduce a class of non-commutative algebras that carry a non-commutative
(geometric) cluster structure which are generated by identical copies of
generalized Weyl algebras. Equivalent conditions for the finiteness of the set
of the cluster variables of these cluster structures are provided. Some
combinatorial data, called \textit{cluster strands,} arising from the cluster
structure are used to construct irreducible representations of generalized Weyl
algebras.
