variables at the nodes are elements of ${\mathcal A}$ and the mutations between nodes are $Q$-system evolutions.

More precisely, we will define the quotient graphs ${\mathcal G}_{{\mathfrak{g}}}$, by identifying nodes corresponding to the same cluster variables and mutation matrices. These are infinite graphs, but they have certain periodicity properties and are therefore easy to describe explicitly.

Our starting goal is to find a graph ${\mathcal G}_{{\mathfrak{g}}}$ for each algebra, and the corresponding cluster variables, which encode the full $Q$-system, in the following sense: The union over all nodes of ${\mathcal G}_{{\mathfrak{g}}}$ of the cluster variables is the full set ${\mathcal A}$.

Of course once the seed $({{\mathbf x}},B)$ is given at any node $t$, then the cluster algebra is defined, that is, the cluster at any other node is defined via the evolution equations. Therefore, our definition gives a cluster algebra on the full tree $\mathbb T_n$. We will have something to say about special properties of the cluster variables on $\mathbb
T_n$ in Section \[poly\], but thus far they have no representation-theoretical interpretation to our knowledge.

This definition of ${\mathcal G}_{{\mathfrak{g}}}$ allows us to encode the $Q$-system inside a cluster algebra. The graphs ${\mathcal G}_{{\mathfrak{g}}}$ which we describe are 