  Recently we discussed how Einstein supergravity tree amplitudes might be
obtained from the original Witten and Berkovits twistor-string theory when
external conformal gravitons are restricted to be Einstein gravitons. Here we
obtain a more systematic understanding of the relationship between conformal
and Einstein gravity amplitudes in that twistor-string theory. We show that
although it does not in general yield Einstein amplitudes, we can nevertheless
obtain some partial twistor-string interpretation of the remarkable formulae
recently been found by Hodges and generalized to all tree amplitudes by Cachazo
and Skinner. The Hodges matrix and its higher degree generalizations encode the
world sheet correlators of the twistor string. These matrices control both
Einstein amplitudes and those of the conformal gravity arising from the Witten
and Berkovits twistor-string. Amplitudes in the latter case arise from products
of the diagonal elements of the generalized Hodges matrices and reduced
determinants give the former. The reduced determinants arise if the
contractions in the worldsheet correlator are restricted to form connected
trees at MHV. The (generalized) Hodges matrices arise as weighted Laplacian
matrices for the graph of possible contractions in the correlators and the
reduced determinants of these weighted Laplacian matrices give the sum of of
the connected tree contributions by an extension of the Matrix-Tree theorem.
