  The tree code for the approximate evaluation of gravitational forces is
extended and substantially accelerated by including mutual cell-cell
interactions. These are computed by a Taylor series in Cartesian coordinates
and in a completely symmetric fashion, such that Newton's third law is
satisfied by construction and hence momentum exactly conserved. The
computational effort is further reduced by exploiting the mutual symmetry of
the interactions. For typical astrophysical problems with N=10^5 and at the
same level of accuracy, the new code is about four times faster than the tree
code. For large N, the computational costs are found to scale almost linearly
with N, which can also be supported by a theoretical argument, and the
advantage over the tree code increases with ever larger N.
