For rooted tree posets, the additive closure of zero-dimensional persistent homology is of finite type, with indecomposables given by reduced rooted tree modules, and decomposition is computable in quadratic time.
Elements of the representation theory of associative algebras
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Decomposing zero-dimensional persistent homology over rooted tree quivers
For rooted tree posets, the additive closure of zero-dimensional persistent homology is of finite type, with indecomposables given by reduced rooted tree modules, and decomposition is computable in quadratic time.