  We introduce a new family of tautological relations of the moduli space of
stable curves of genus $g$. These relations are obtained by computing the
Poincar\'e-dual class of empty loci in the Hodge bundle. We use these relations
to obtain a new expression for the Chern classes of the Hodge bundle. We prove
that the $(g-i)$th class can be expressed as a linear combination of
tautological classes involving only stable graphs with at most $i$ loops. In
particular the top Chern class may be expressed with trees. This property was
expected as a consequence of the DR/DZ equivalence conjecture by
Buryak-Gu\'er\'e-Rossi.
