This paper expands on recent studies of very large treebank collections aiming to find empirical evidence for language universals, specifically for the functionally motivated Dependency Length Minimization (DLM) hypothesis. According to DLM grammars are set up to support the expression of utterances in a way that minimizes the distance between heads and dependents. We construct several incremental baselines that lead from the random free order linearization to the real language by adding various word order constraints. We conduct detailed analyses on 55 treebanks and find that all of the constraints contribute to DLM. We show that DLM on the one hand shapes the regularity and on the other motivates the attested exceptions from canonical word order. The findings contribute to a more fine-grained, differentiated picture of the role of DLM in the interaction of competing constraints on grammar and language use.