The idea of articulating a propositional logic that adds a relevance-sensitive implication connective to the usual truth-functional ones has been approached in many ways, e.g. using axioms and derivation rules, natural deduction systems, possible worlds or states equipped with relations or operations, algebraic structures, consecution systems etc. None are very satisfactory. In the 1970s, semantic decomposition trees (aka truth-trees, semantic tableaux, analytic tableaux) were briefly considered, but they did not get far and were swept away by the tsunami of Routley/Meyer possible worlds with ternary relations. We renew that approach using a notion of parity for the branches of a tree or, alternatively, an even more global one of sibling-parity for the entire tree. There are some nice results and, above all, some big open questions.