> more trees today than ever in history?
Depending on how big you count history this sounds really implausible.
> carbon is what plants eat.
A slogan commonly repeated by global warming denialists, because it's true but highly misleading. Plants also primarily require water, and the temperature rise dries out a lot of places.
History := the recorded past (in this case I mean the last few hundred years)
> Desertification is currently increasing
Okay, you're right about this based on the link you sent. In my head I was considering arctic regions / tundra as desert as well, as their recent forestation rate is much faster.
> temperature rise dries out a lot of places
This is true but it also causes other places to become more humid. Higher temperature climate has more liquid water + more entropy ==> more active water cycle on global average. (This is a gross oversimplification but my point is that the increase in temperature is an increase in chaos, and so water that is currently frozen somewhere will be moving around.)
The important idea to note here is that we can't just ask "does global warming cause more X"? Because the answer can be yes in some areas and no in others, and an increase in X somewhere does not mean a net increase globally.