  We give an algebraic construction of the topological graph-tree configuration
pairing of Sinha and Walter beginning with the classical presentation of Lie
coalgebras via coefficients of words in the associative Lie polynomial. Our
work moves from associative algebras to preLie algebras to graph complexes,
justifying the use of graph generators for Lie coalgebras by iteratively
expanding the set of generators until the set of relations collapses to two
simple local expressions. Our focus is on new computational methods allowed by
this framework and the efficiency of the graph presentation in proofs and
calculus involving free Lie algebras and coalgebras. This outlines a new way of
understanding and calculating with Lie algebras arising from the graph
presentation of Lie coalgebras.
