There is a formula for the Laurent expansions of cluster variables in terms of matchings in "snake graphs" for cluster algebras from any surface. For the question type $A$ (a disk with marked points on the boundary) and type $D$ (a once punctured disk with marked points on the boundary) come from surfaces.

[Positivity for cluster algebras from surfaces
][1] by Musiker, Schiffler, and  Williams has this for any surface. [Cluster expansion formulas and perfect matchings][2] by Musiker and Schiffler has treatment for unpunctured surfaces. These [notes][3] of Schiffler also treat cluster algebras from surfaces including the Laurent expansion. Canakci and Schiffler also have a series of paper on various aspects of snake graphs.


  [1]: https://arxiv.org/abs/0906.0748
  [2]: https://arxiv.org/abs/0810.3638
  [3]: https://webusers.imj-prg.fr/~patrick.le-meur/CIMPA/Course-Schiffler.pdf