Characterization of hierarchically clustered graphs

There is a strong relationship between (rooted) trees and hierarchically clustered graphs as can be seen here:

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The tree – consisting of circles (green = the root, blue, red, black = the leaves) and thin lines – can be seen as the result of a hierarchical cluster analysis of the graph consisting of the black circles and thick lines. Hierarchically clustered graphs can be defined as those graphs for which an appropriately defined hierarchical cluster analysis yields a (non-trivial) tree.

Trees are very elegantly defined and characterized as graphs that are connected and have no cycles. But how can their companions – the hierarchically clustered graphs – be characterized in a similar elegant way, or characterized at all (next to their definition above)?

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