Labeled Trees and the Algebra of Differential Operators

This paper is concerned with the effective symbolic computation of operators under composition. Examples include differential operators under composition and vector fields under the Lie bracket. A basic fact about such operators is that, in general, they do not commute. A basic fact about applied mathematicians is that they often rewrite expressions involving noncommuting operators in terms of other operators which do commute. If the original expression enjoys a certain symmetry, then the naive rewriting requires the computation of terms which in the end cancel. In this paper we analyse data structures consisting of formal linear combinations of rooted labeled trees. We define a multiplication on rooted labeled trees, thereby making the set of these data structures into an associative algebra. We then define an algebra homomorphism from the original algebra of operators into this algebra of trees. The cancellation which occurs