  This paper extends the key concept of persistence within Topological Data
Analysis (TDA) in a new direction. TDA quantifies topological shapes hidden in
unorganized data such as clouds of unordered points. In the 0-dimensional case
the distance-based persistence is determined by a single-linkage (SL)
clustering of a finite set in a metric space. Equivalently, the 0D persistence
captures only edge-lengths of a Minimum Spanning Tree (MST). Both SL dendrogram
and MST are unstable under perturbations of points. We define the new
stable-under-noise mergegram, which outperforms previous isometry invariants on
a classification of point clouds by PersLay.
