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A systematic study of the method of regions is crucial for evaluating Feynman Integrals in asymptotic limits. In this paper, we focus on expansions in Euclidean signature, where the regions can be formulated in terms of subgraphs, called the expansion by subgraph technique. We establish an "asymptotic Hopf algebra" which can be used to formulate the expansion by subgraph in terms of motic subgraphs. Explorations into the Hopf Algebraic structure reveals striking similarities to the well-known Renormalisation Hopf Algebra by Connes and Kreimer, indicating deeper relations between renormalisation and asymptotic expansions. We also introduce a novel Hopf monoid formulation as a future direction, and discuss how it is possibly a more fundamental structure than the Hopf Algebra. This paper lays down the Hopf algebraic foundation based on which a rigorous and systematic exploration into the method of regions and renormalisation can be expected in the future.