Forest handles the splines in 2D, so by default it doesn't create overlapped items. There are two ways to fix it, both using surfaces:
a) Using XY Surfaces:
- Create one surface for each spline (for example, applying a UVW map modier), and assign them to Forest (Surface rollout)
Note: You must create several surfaces, because in this mode a same surface object cannot have overlapped faces.
b) Using UV Surfaces:
- Same as before, but there are not restrictions with the overlapped faces. So, you can attach all the surfaces in a unique mesh.