A new logic for reasoning over finite trees is proposed. This logic currently offers the best balance known between expressivity and complexity for decidability. It is as expressive as monadic second-order logic whereas its satisfiability is shown decidable in time complexity 2^O(n) w.r.t. size n of the formula. We present an effective algorithm in practice using symbolic techniques (BDDs), and use it for the static analysis of XPath queries in the presence of regular tree types, including XPath typing.