The universal homogeneous binary tree

A partial order is called semilinear iff the upper bounds of each element are linearly ordered and any two elements have a common upper bound. There exists, up to isomorphism, a unique countable semilinear order which is dense, unbounded, binary branching, and without joins, which we denote by (S2;≤). We study the reducts of (S2;≤), that is, the relational structures with domain S2, all of whose relations are first-order definable in (S2;≤). Our main result is a classification of the model-complete cores of the reducts of S2. From this, we also obtain a classification of reducts up to first-order interdefinability, which is equivalent to a classification of all closed permutation groups that contain the automorphism group of (S2;≤).