From playing, it looks to me that it's not a Savanna only problem, in df2014 trees simply grow fully too fast and if you don't regularly cut as much as you can, in a few years you always end impenetrable jungles, very likely impacting negatively your game speed too.
Tough i have not tried desert biome, i wonder if the problem can still happen there if you left trees unchecked.
There was a bug report about it that Toady marked as fixed for 40.05 :
http://www.bay12games.com/dwarves/mantisbt/view.php?id=7581
I don't even want to imagine how it was in 40.04 if what i see now is supposed to be working as intended
If I remember correctly, in 40.04 trees reached maturity after only one season, or something crazy like that. In general, Toady marking something as fixed doesn't mean that it works perfectly, just that its not ridiculously broken...
I often embark on biomes with lots of trees, and I haven't noticed the trees getting thicker in regions where I don't cut them down. Once they reach a spacing of about 4 tiles between them, no more appear to grow. I doubt this is an accident, as this is the spacing needed to allow wagons to pass through. I also haven't noticed my FPS suffering badly, and my computer is pretty slow.
In other words, while it's of course not realistic, I don't think there's any harm to letting trees grow to maximum density, other than providing a route for invaders to climb over your walls, and the crashes which occur if they grow on top of your constructions.
If you are worried about these issues, you can periodically designate unpaved roads in regions where you don't want trees, and a dwarf will very quickly remove all saplings (and other plants) from these areas. If you want to make this solution permanent, you can make a paved road, which can be made of wood, which you must have in excess if you are complaining about trees. Paved roads next to any above ground constructions is probably a good idea to avoid tree-related crashes.