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We compute the domination monoid in the theory    DMT  $\mathsf {DMT}$    of dense meet‐trees. In order to show that this monoid is well‐defined, we prove  weak binarity  of    DMT  $\mathsf {DMT}$    and, more generally, of certain expansions of it by binary relations on sets of open cones, a special case being the theory    DTR  $\mathsf {DTR}$    from [7]. We then describe the domination monoids of such expansions in terms of those of the expanding relations.

Weakly binary expansions of dense meet‐trees