First, this talk provides an overview of the topological classification of Hamiltonian flows on surfaces, which serve as a mathematical model for two-dimensional incompressible and viscous flows. Next, building upon this theory, we extend to the torus the framework for converting the topology of any Hamiltonian flow on a sphere into a rooted planar tree, called a partially Cyclically-Ordered rooted Tree (COT), along with its string expression (COT representation). Applying this conversion algorithm to snapshots of enstrophy cascade turbulence and inverse energy cascade turbulence, we demonstrate that complicated flow patterns can be effectively represented as trees and letter sequences.