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

Abstract Consider any locally checkable labeling problem $$\Pi $$ Π in rooted regular trees : there is a finite set of labels $$\Sigma Σ , and for each label $$x \in \Sigma x ∈ we specify what are permitted combinations the children an ...