We present a family of new top-down parsing algorithms for arbitrary context-free grammars based on the notion of a "tree of possible left-derivations". When coded in a special way, this tree can be compressed to a graph. While the time complexity on uniform RAMs is O(n^3) in worst case, we need O(n^2) space. We show that the asymptotic time complexities are equal to Earley's algorithms, and give some arguments to show our algorithm will be better in almost all cases.