  We study the decomposition of zero-dimensional persistence modules, viewed as
functors valued in the category of vector spaces factorizing through sets.
Instead of working directly at the level of vector spaces, we take a step back
and first study the decomposition problem at the level of sets.
  This approach allows us to define the combinatorial notion of rooted subsets.
In the case of a filtered metric space $M$, rooted subsets relate the
clustering behavior of the points of $M$ with the decomposition of the
associated persistence module. In particular, we can identify intervals in such
a decomposition quickly. In addition, rooted subsets can be understood as a
generalization of the elder rule, and are also related to the notion of
constant conqueror of Cai, Kim, M\'emoli and Wang. As an application, we give a
lower bound on the number of intervals that we can expect in the decomposition
of zero-dimensional persistence modules of a density-Rips filtration in
Euclidean space: in the limit, and under very general circumstances, we can
expect that at least 25% of the indecomposable summands are interval modules.
