  A functor from the category of directed trees with inclusions to the category
of commutative C*-algebras with injective *-homomorphisms is constructed. This
is used to define a functor from the category of directed graphs with
inclusions to the category of C*-algebras with injective *-homomorphisms. The
resulting C*-algebras are identified as Toeplitz graph algebras. Graph algebras
are proved to have inductive limit decompositions over any family of subgraphs
with union equal to the whole graph. The construction is used to prove various
structural properties of graph algebras.
