  LP algebras, introduced by Lam and Pylyavskyy, are a generalization of
cluster algebras. These algebras are known to have the Laurent phenomenon, but
positivity remains conjectural. Graph LP algebras are finite LP algebras
encoded by a graph. For the graph LP algebra defined by a tree, we define a
family of clusters called rooted clusters. We prove positivity for these
clusters by giving explicit formulas for each cluster variable. We also give a
combinatorial interpretation for these expansions using a generalization of
$T$-paths.
