  We study a basis of the polynomial ring that we call forest polynomials. This
family of polynomials is indexed by a combinatorial structure called indexed
forests and permits several definitions, one of which involves flagged
P-partitions. As such, these polynomials have a positive expansion in the basis
of slide polynomials. By a novel insertion procedure that may be viewed as a
generalization of the Sylvester correspondence we establish that Schubert
polynomials decompose positively in terms of forest polynomials. Our insertion
procedure involves a correspondence on words which allows us to show that
forest polynomials multiply positively.
  We proceed to show that forest polynomials are a particularly convenient
basis in regards to studying the quotient of the polynomial ring modulo the
ideal of positive degree quasisymmetric polynomials. This aspect allows us to
give a manifestly nonnegative integral description for the Schubert class
expansion of the cohomology class of the permutahedral variety in terms of a
parking procedure. We study the associated combinatorics in depth and introduce
a multivariate extension of mixed Eulerian numbers.
