Dependency parsing algorithms capable of producing the types of crossing dependencies seen in natural language sentences have traditionally been orders of magnitude slower than algorithms for projective trees. For 95.8–99.8% of dependency parses in various natural language treebanks, whenever an edge is crossed, the edges that cross it all have a common vertex. The optimal dependency tree that satisfies this 1-Endpoint-Crossing property can be found with an O(n4) parsing algorithm that recursively combines forests over intervals with one exterior point. 1-Endpoint-Crossing trees also have natural connections to linguistics and another class of graphs that has been studied in NLP.