  This paper addresses several structural aspects of the insertion-elimination
algebra, a Lie algebra that can be realized in terms of tree-inserting and
tree-eliminating operations on the set of rooted trees. In particular, we
determine the finite-dimensional subalgebras, the automorphism group, the
derivation group, and a generating set for the insertion-elimination Lie
algebra. Many parts of the results are stated for a more general class of Lie
algebras and reproduce results for the generalized Virasoro algebras.
