On some varieties associated with trees

This article considers some affine algebraic varieties attached to finite trees and closely related to cluster algebras. Their definition involves a canonical coloring of vertices of trees into three colors. These varieties are proved to be smooth and to admit sometimes free actions of algebraic tori. Some results are obtained on their number of points over finite fields and on their cohomology. Contents 1 Combinatorics of trees 4 1.1 Canonical red-orange-green coloring of trees............ 4 1.2 Further properties of the coloring.................. 5 2 Affine algebraic varieties