Foams are surfaces with branch lines at which three sheets merge. They have been used in the categorification of $\mathrm{sl}(3)$ quantum knot invariants and also in physics. The $2D$-TQFT of surfaces, on the other hand, is classified by means of commutative Frobenius algebras, where saddle points correspond to multiplication and comultiplication. In this paper, we explore algebraic operations that branch lines derive under TQFT. In particular, we investigate Lie bracket and bialgebra structures. Relations to the original Frobenius algebra structures are discussed both algebraically and diagrammatically.