I do not yet know whether the forest is anything but cosmetic, but from the land value map it looks like the trees do not do much of anything, although in SimCity 4 wild vegetation was an important tool for increasing desirability.
To plant or cut-down the forest one needs to select Parks > Nature (§§) then scroll to the right where they would find the tools for the job, as seen on the screenshot below.
There is no immediate reaction to the use of the tool (except for the money leaving one’s account in an instant, much like EA pre-order), but I suppose it grows over time (much like Origin’s delivery of pre-ordered titles on release day). That said, try as I may, I had not had any tree show up anywhere I had planted, so this tool still needs figuring out.
The above quote is wrong, trees do show up immediately, but not always. Still do not know if it is a feature or a bug.
Apologies for strange and foreign tongue as per interface. I am yet to figure out how to change it for demonstration purposes.