  In this paper we prove that if we consider the standard real metric on
simplicial rooted trees then the category Tower-Set of inverse sequences can be
described by means of the bounded coarse geometry of the naturally associated
trees. Using this we give a geometrical characterization of Mittag-Leffler
property in inverse sequences in terms of the Lipschitz, and metrically proper,
homotopy type of the corresponding tree and its maximal geodesically complete
subtree. We also some consequences in shape theory, in particular we obtain
some new representations of shape morphisms related to infinite brunches in
trees.
