The cake is lovely and I'm curious how you all think the peeled birch was made? Wafer paper? 

I think it's probably wafer paper and gumpaste flowers, but it's hard to tell because it's so small. It could be fondant.

Personally, I'm not a fan of thick layers of wafer paper on the outside of a cake because it hardens up if it gets wet and dries, and it tastes like glue. If I was going to do this cake I'd probably use fondant with white candy clay in it, or just candy clay rolled out really thin. IT would taste better, anyway.

Fondant works great for this effect for the very qualities of it that drive people crazy: tearing and cracking.  Roll it thin, slap it on lazily, paint on some darker accents, you've got bark!

this one does look really thin like wafer paper, as cc said, but I think you could get the same effect with fondant, just roll it really thin and thin the edges before applying.