Characterizing the distribution of crossing dependencies in natural language dependency trees is a crucial task for building parsers and understanding the formal properties of human language. A number of formal restrictions on crossing dependencies have been proposed, including bounds on gap degree, edge degree, and end-point crossings. Here we ask whether the empirical distribution of crossing dependencies in dependency treebanks offers evidence for these formal restrictions as true, independent constraints on dependency trees, or whether the distribution can be explained using other, more generic constraints affecting dependency trees. Specifically, we explore the null hypothesis that crossing dependencies are formally unrestricted, but occur at a low rate. We implement the null hypothesis using random trees where crossing dependencies occur at the same rate as in natural language trees, but without any formal restrictions. We find that this baseline generally does not reproduce the same distribution of gap degree, edge degree, endpoint-crossing, and heads' depth difference as real trees, suggesting that these formal constraints are a consequence of factors beyond the rate of crossing dependencies alone.