We study the functorial characterisation of bisimulation-based equivalences over a categorical model of labelled trees. We show that in a setting where all labels are visible, strong bisimilarity can be characterised in terms of enriched functors by relying on the reflection of paths with their factorisations. For an enriched functor F, this notion requires that a path (an internal morphism in our framework) π going from F(A) to C corresponds to a path p going from A to K, with F(K) = C, such that every possible factorisation of π can be lifted in an appropriate factorisation of p. This last property corresponds to a Conduché property for enriched functors, and a very rigid formulation of it has been used by Lawvere to characterise the determinacy of physical systems. We also consider the setting where some labels are not visible, and provide characterisations for weak and branching bisimilarity. Both equivalences are still characterised in terms of enriched functors that reflect paths with their factorisations: for branching bisimilarity, the property is the same as the one used to characterise strong bisimilarity when all labels are visible; for weak bisimilarity, a weaker form of path factorisation lifting is needed. This fact can be seen as evidence that strong and branching bisimilarity are strictly related and that, unlike weak bisimilarity, they preserve process determinacy in the sense of Milner.