  We study the existence and structure of branch points in two-phase free
boundary problems. More precisely, we construct a family of minimizers to an
Alt- Caffarelli-Friedman type functional whose free boundaries contain branch
points in the strict interior of the domain. We also give an example showing
that branch points in the free boundary of almost-minimizers of the same
functional can have very little structure. This last example stands in contrast
with recent results of De Philippis- Spolaor-Velichkov on the structure of
branch points in the free boundary of stationary solutions.
