  This paper establishes a connection between binary subwords and perfect
matchings of a snake graph, an important tool in the theory of cluster
algebras. Every binary expansion w can be associated to a piecewise-linear
poset P and a snake graph G. We construct a tree structure called the antichain
trie which is isomorphic to the trie of subwords introduced by Leroy, Rigo, and
Stipulanti. We then present bijections from the subwords of w to the antichains
of P and to the perfect matchings of G.
