A $\mathbb{Z}$-basis for the cluster algebra associated to an affine quiver

The canonical bases of cluster algebras of finite types and rank 2 are given explicitly in \cite{CK2005} and \cite{SZ} respectively. In this paper, we will deduce $\mathbb{Z}$-bases for cluster algebras for affine types ${\tilde{A}}_{n,n},\tilde{D}$ and $\tilde{E}$. Moreover, we give an inductive formula for computing the multiplication between two generalized cluster variables associated to objects in a tube.