  We study cluster algebras with principal coefficient systems that are
associated to unpunctured surfaces. We give a direct formula for the Laurent
polynomial expansion of cluster variables in these cluster algebras in terms of
perfect matchings of a certain graph $G_{T,\gamma}$ that is constructed from
the surface by recursive glueing of elementary pieces that we call tiles. We
also give a second formula for these Laurent polynomial expansions in terms of
subgraphs of the graph $G_{T,\gamma}$.
