  In this expository paper, we present a construction of tree modules and
combine it with (infinite dimensional) tilting theory and relative
Mittag-Leffler conditions in order to explore limits of the approximation
theory of modules. We also present a recent generalization of this construction
due to Saroch which applies to factorization properties of maps, and yields a
solution of an old problem by Auslander concerning existence of almost split
sequences.
