Alright, I was able to duplicate your exact problem shown in the video you posted. Its definitely a graphics problem your having.
Go to "Options" in the upper right hand corner of your Simcity in game menu. Scroll up to "Settings" then scroll down to "Geometry"
Geometry needs to be at least on a setting of Medium to view forests.
When I reduced it to Low I duplicated your exact same results with only a couple of trees showing no matter how many clicks I gave it.
When I Set it to Medium or High I was able to see planted forest. I tried every combination in the graphics settings the "Geometry" setting was the only one that affected the number of trees viewed after planting.
To sum it up you need to have a graphics card capable of medium on the geometry setting to view planted forest.
Please check this out and tag this solved if it helps you so other players will know.
