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Let ( $\rm L$ ;C) be the (up to isomorphism unique) countable homogeneous structure carrying a binary branching C-relation. We study the reducts of ( $\rm L$ ;C), i.e., the structures with domain $\rm L$ that are first-order definable in ( $\rm L$ ;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 ( $\rm L$ ;C). We also study the endomorphism monoids of such reducts and show that they fall into four categories.



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The Journal of Symbolic Logic
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