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Let $\Delta \subset {𝐑}^2$ be an integral convex polygon. G. Mikhalkin introduced the notion of, a class of real algebraic curves, defined by polynomials supported on $\Delta$ and contained in the corresponding toric surface. He proved their existence, viamethod, and that the topological type of their real parts is unique (and determined by $\Delta$). This paper is concerned with the description of the analogous statement in the case of a smoothing of a real plane branch $(C,0)$. We introduce the class ofsmoothings of $(C,0)$ by passing through...