  The syntactic structure of a sentence can be modeled as a tree where vertices
are words and edges indicate syntactic dependencies between words. It is
well-known that those edges normally do not cross when drawn over the sentence.
Here a new null hypothesis for the number of edge crossings of a sentence is
presented. That null hypothesis takes into account the length of the pair of
edges that may cross and predicts the relative number of crossings in random
trees with a small error, suggesting that a ban of crossings or a principle of
minimization of crossings are not needed in general to explain the origins of
non-crossing dependencies. Our work paves the way for more powerful null
hypotheses to investigate the origins of non-crossing dependencies in nature.
