The Reducts of the homogeneous Binary Branching C-Relation

Let (L;C ) be the (up to isomorphism unique) countable homogeneous structure carrying a binary branching C-relation. We study the reducts of (L;C ), i.e., the structures with domain L that are first-order definable in (L;C ). We show that up to existential interdefinability, there are finitely many such reducts. This implies that there are finitely many reducts up to first-order interdefinability, thus confirming a conjecture of Simon Thomas for the special case of (L;C ). We also study the endomorphism monoids of such reducts and show that they fall into four categories. §