  We give a description of the structure of finite Morse index solutions to two
free boundary problems in $\mathbb{R}^2$. These free boundary problems are
models of phase transition and they are closely related to minimal
hypersurfaces. We show that these finite Morse index solutions have finitely
many ends and they converge exponentially to these ends at infinity. As an
important tool in the proof, a quadratic decay estimate for the curvature of
free boundaries is established.
