  We begin by considering the graded vector space with a basis consisting of
rooted trees, graded by the count of non-root vertices. We define two linear
operators on this vector space, the growth and pruning operators, which
respectively raise and lower grading; their commutator is the operator that
multiplies a rooted tree by its number of vertices. We define an inner product
with respect to which the growth and pruning operators are adjoint, and obtain
several results about the multiplicities associated with each operator.
  The symmetric algebra on the vector space of rooted trees (after a degree
shift) can be endowed with a coproduct to make a Hopf algebra; this was defined
by Kreimer in connection with renormalization. We extend the growth and pruning
operators, as well as the inner product mentioned above, to Kreimer's Hopf
algebra. On the other hand, the vector space of rooted trees itself can be
given a noncommutative multiplication: with an appropriate coproduct, this
gives the Hopf algebra of Grossman and Larson. We show the inner product on
rooted trees leads to an isomorphism of the Grossman-Larson Hopf algebra with
the graded dual of Kreimer's Hopf algebra, correcting an earlier result of
Panaite.
