Linear Context-Free Rewriting System (LCFRS) is an extension of Context-Free Grammar (CFG) in which a non-terminal can dominate more than a single continu-ous span of terminals. Probabilistic LCFRS have recently successfully been used for the direct data-driven parsing of discontin-uous structures. In this paper we present a parser for binary PLCFRS of fan-out two, together with a novel monotonous estimate for A ∗ parsing, with which we conduct ex-periments on modified versions of the Ger-man NeGra treebank and the Discontinuous Penn Treebank in which all trees have block degree two. The experiments show that compared to previous work, our approach provides an enormous speed-up while de-livering an output of comparable richness. 1