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Using the ordered analogue of Farley-Sabalka's discrete gradient field on theconfiguration space of a graph, we unravel a levelwise behavior of thegenerators of the pure braid group on a tree. This allows us to generalizeFarber's equivariant description of the homotopy type of the configurationspace on a tree on two particles. The results are applied to the calculation ofall the higher topological complexities of ordered configuration spaces ontrees on any number of particles.

Farley-Sabalka's Morse-theory model and the higher topological complexity of ordered configuration spaces on trees