Dendrograms for persistence diagrams

When I am explaining persistence to Engineers, I start from dendrograms for hierarchical clustering, as most of them are familiar with that concept, and then present persistence diagrams and barcodes as analogies for "higher order topological properties".

I just realized that I had the wrong impression (because I never thought about it) that 0-persistence as a special case, has some properties which makes dendrograms possible as opposed their "coarser" counterparts, barcodes and persistence diagrams, for higher persistence. But, this is only because we implicitly assume a canonical basis for \( C_0 \) in single linkage clustering, or 0-persistence.

It is not always the case that a natural canonical basis (which makes some sort of sense) exists for higher dimensional chain spaces. However, in the case of Rips filtration based on Euclidean metric for point clouds, especially in \(\mathbb{R}^2\) and \(\mathbb{R}^3\), we can use the Alexander duality to define a canonical basis, and thus actually obtain a dendrogram. More generally, whenever we have a meaningful canonical basis for our chain spaces, we can have dendrograms for persistence.