Hierarchical clustering is a popular method for analyzing data which\nassociates a tree to a dataset. Hartigan consistency has been used extensively\nas a framework to analyze such clustering algorithms from a statistical point\nof view. Still, as we show in the paper, a tree which is Hartigan consistent\nwith a given density can look very different than the correct limit tree.\nSpecifically, Hartigan consistency permits two types of undesirable\nconfigurations which we term over-segmentation and improper nesting. Moreover,\nHartigan consistency is a limit property and does not directly quantify\ndifference between trees.\n In this paper we identify two limit properties, separation and minimality,\nwhich address both over-segmentation and improper nesting and together imply\n(but are not implied by) Hartigan consistency. We proceed to introduce a merge\ndistortion metric between hierarchical clusterings and show that convergence in\nour distance implies both separation and minimality. We also prove that uniform\nseparation and minimality imply convergence in the merge distortion metric.\nFurthermore, we show that our merge distortion metric is stable under\nperturbations of the density.\n Finally, we demonstrate applicability of these concepts by proving\nconvergence results for two clustering algorithms. First, we show convergence\n(and hence separation and minimality) of the recent robust single linkage\nalgorithm of Chaudhuri and Dasgupta (2010). Second, we provide convergence\nresults on manifolds for topological split tree clustering.\n