  We present a new formula for the antipode of incidence Hopf algebras. This
formula is expressed as an alternating sum over forests. First, we prove the
formula for incidence Hopf algebras of families of lattices by exhibiting a map
from chains of a lattice to forests. Then, we extend the definition and present
an analogous formula for the antipode of incidence Hopf algebras of families of
posets. We characterize those families for which our formula is
cancellation-free.
