Would a melted crystal recrystallize back into the same lattice structure when frozen?
If cooling is rapid enough, then crystals don't have time to form, so the substance may be amorphous or glassy (though there may be some localized structure). For example, there are metallic glasses and amorphous water ice. Metallic glasses may have improved wear and corrosion resistance over a crystalline alloy. One interesting intermediate form is a quasicrysal, which can have 5-fold symmetry. Roger Penrose proposed this geometrical tiling before it was found in metallic glasses.
It depends on the conditions in which the material freezes. For example, graphite can be forced into a diamond structure by heat and pressure. A pseudomorph could form if the substrate initializes crystallization in a different form.