Before revision:  It turns out that multiple ancestral processes are associated to the same forward dynamics. We characterise the set of optimal ancestral structures and provide a recipe to construct them from the drift. In particular, this allows us to recover well-known ancestral structures of the literature .
After revision:  It turns out that multiple ancestral processes are associated to the same forward process is compatible with multiple backward models. In this context we introduce suitable notions for minimality among the ancestral processes and characterise the corresponding parameter sets. In this way we recover classic ancestral structures as minimal ones .