We introduce gap inheritance, a new structural property on trees, which provides a way to quantify the degree to which intervals of descendants can be nested. Based on this property, two new classes of trees are derived that provide a closer approximation to the set of plausible natural language dependency trees than some alternative classes of trees: unlike projective trees, a word can have descendants in more than one interval; unlike spanning trees, these intervals cannot be nested in arbitrary ways. The 1-Inherit class of trees has exactly the same empirical coverage of natural language sentences as the class of mildly non-projective trees, yet the optimal scoring tree can be found in an order of magnitude less time. Gap-minding trees (the second class) have the property that all edges into an interval of descendants come from the same node, and thus an algorithm which uses only single intervals can produce trees in which a node has descendants in multiple intervals.
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