  We compute the domination monoid in the theory DMT of dense meet-trees. In
order to show that this monoid is well-defined, we prove weak binarity of DMT
and, more generally, of certain expansions of it by binary relations on sets of
open cones, a special case being the theory DTR from arXiv:1909.04626. We then
describe the domination monoids of such expansions in terms of those of the
expanding relations.
