This paper is centered around two main contributions: the first one consists in introducing several procedures for generating random dependency trees with constraints; we later use these artificial trees to compare their properties with the properties of natural trees (i.e trees extracted from treebanks) and analyze the relationships between these properties in natural and artificial settings in order to find out which relationships are formally constrained and which are linguistically motivated. We take into consideration five metrics: tree length, height, maximum arity, mean dependency distance and mean flux weight, and also look into the distribution of local configurations of nodes. This analysis is based on UD treebanks (version 2.3, Nivre et al. 2018) for four languages: Chinese, English, French and Japanese.