The component-tree is a popular data structure, involved in grey-level image modelling and processing. Basically, a component-tree can be seen as the Hasse diagram of the inclusion relation for the threshold-sets of a non-directed graph, which vertices are endowed with totally ordered values. In this talk, we discuss about various ways of extending the component-tree beyond its current hypotheses (from non-directed to directed graphs; from total to partial orders on values; ...), with the purpose to design new data structures, with a wider applicative range. We present recent results on these topics, but also the remaining challenges to be tackled.