I don't use AtoC or AA at all and get the same; but only when looking at those specific trees. Usually my FPS drops 10+ frames into the mid-20's.
I think it seems worse in prone because your view is more stable and can focus on the tree instead of swaying around, so I don't think they're directly related.
Arma 2 had a similar problem with certain trees...they just require more resources to render. Hopefully they can optimize it