The construction of a theory for binary trees is presented, based on the systematic exploration of the properties necessary for the proof-based synthesis and certification of sorting algorithms for binary trees. The process is computer supported, being realised in the frame of the Theorema system, with some additional proofs in Coq required for algorithm certification. The result of the exploration consists in 11 definitions, 3 axioms, and more than 200 properties. Also, more than 5 algorithms for sorting binary trees are generated.