Tree polynomials and non-associative Gröbner bases

In this article the basic notions of a theory of Gröbner bases for ideals in the non-associative, non-commutative algebra K{X} with a unit freely generated by a set X over a field K are discussed. The monomials in this algebra can be identified with the set of isomorphism classes of X-labelled finite, planar binary rooted trees where X is the set of free algebra generators. The elements of K{X} are called tree polynomials. We describe a criterion for a system of polynomials to constitute a Gröbner basis. It can be seen as a non-associative version of the Buchberger criterion. A formula is obtained for the generating series of a reduced Gröbner basis for the ideal of non-associative and non-commutative relations of an algebra relative to a system of algebra generators and an admissible order on the monomials. If the algebra is graded it specializes to a general Hilbert series formula in terms of generators and relations. We also report on new results concerning non-associative power series like exp,log and the Hausdorff series and on problems related to Hopf algebras of trees. Reduced Gröbner bases for closed ideals in tree power series algebras K{{X}} are considered.