  We develop a quasisymmetric analogue of the combinatorial theory of Schubert
polynomials and the associated divided difference operators. Our counterparts
are "forest polynomials", and a new family of linear operators, whose theory of
compositions is governed by forests and the "Thompson monoid". Our approach
extends naturally to $m$-colored quasisymmetric functions.
  We then give several applications of our theory to fundamental quasisymmetric
functions, the study of quasisymmetric coinvariant rings and their associated
harmonics, and positivity results for various expansions. In particular we
resolve a conjecture of Aval-Bergeron-Li regarding quasisymmetric harmonics.
