We present a new syntactic parser that works left-to-right and top down, thus maintaining a fully-connected parse tree for a few alternative parse hypotheses. All of the commonly used statistical parsers use context-free dynamic programming algorithms and as such work bottom up on the entire sentence. Thus they only find a complete fully connected parse at the very end. In contrast, both subjective and experimental evidence show that people understand a sentence word-to-word as they go along, or close to it. The constraint that the parser keeps one or more fully connected syntactic trees is intended to operationalize this cognitive fact. Our parser achieves a new best result for topdown parsers of 89.4%,a 20% error reduction over the previous single-parser best result for parsers of this type of 86.8% (Roark, 2001). The improved performance is due to embracing the very large feature set available in exchange for giving up dynamic programming.