  We extend the concept of a Hubbard tree, well established and useful in the
theory of polynomial dynamics, to the dynamics of transcendental entire
functions. We show that Hubbard trees in the strict traditional sense, as
invariant compact trees embedded in $\mathbb{C}$, do not exist even for
post-singularly finite exponential maps; the difficulty lies in the existence
of asymptotic values. We therefore introduce the concept of a Homotopy Hubbard
Tree that takes care of these difficulties.
  Specifically for the family of exponential maps, we show that every
post-singularly finite map has a Homotopy Hubbard tree that is unique up to
homotopy, and we show that post-singulary finite exponential maps can be
classified in terms of Homotopy Hubbard Trees, using a transcendental analogue
of Thurston's topological characterization theorem of rational maps.
