  We develop a canonical pairing between trees and graphs, which passes to
their quotients by Jacobi identities. This pairing is an effective and simple
tool for understanding the Lie and Poisson operads, providing canonical duals.
In the course of showing that this pairing is perfect we reprove some standard
facts about the modules Lie(n), establishing standard bases as well as giving a
new means to reduce to those bases. We then move on to define duals to free Lie
algebras and to develop product, coproduct and operad structures. We give a
brief account here to be built on in a number of different directions in future
work.
