  This paper shows that the topological structures of particle orbits generated
by a generic class of vector fields on spherical surfaces, called {\it the flow
of finite type}, are in one-to-one correspondence with discrete structures such
as trees/graphs and sequence of letters. The flow of finite type is an
extension of structurally stable Hamiltonian vector fields, which appear in
many theoretical and numerical investigations of 2D incompressible fluid flows.
Moreover, it contains compressible 2D vector fields such as the Morse--Smale
vector fields and the projection of 3D vector fields onto 2D sections. The
discrete representation is not only a simple symbolic identifier for the
topological structure of complex flows, but it also gives rise to a new
methodology of topological data analysis for flows when applied to data brought
by measurements, experiments, and numerical simulations of complex flows. As a
proof of concept, we provide some applications of the representation theory to
2D compressible vector fields and a 3D vector field arising in an industrial
problem.
