A partial order is called semilinear if 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 existentially closed semilinear order, which we denote by (S₂;≤). We study the reducts of (S₂;≤), i.e. 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 subgroups of the full symmetric group on S₂ that contain the automorphism group of (S₂;≤) and are closed with respect to the pointwise convergence topology.