Property FA is not a profinite property

We exhibit infinitely many pairs of non-isomorphic finitely presented, residually finite groups $\mathrm{\Delta }$ and $\mathrm{\Gamma }$ with $\mathrm{\Delta }$ having Property FA, $\mathrm{\Gamma }$ having a non-trivial action on a tree and $\mathrm{\Delta }$ and $\mathrm{\Gamma }$ having isomorphic profinite completions.