An amazing desolate area, that feels like it truly deserves the label "moonscape".
You can camp near pretty much any lake, and you have a lot of choices. You can travel the basin point-to-point, as it is reasonably level, Humphreys provides orientation, and the terrain, until you are on a mountain/rustic pass, is benign.
Not sure where you are going to find a tree at that elevation.
I vaguely remember some I camped in at one of the Golden Trout Lakes, but there are sure not many in the basin, proper.
I've been in the basin in major weather/lightning, and did not feel at risk, with lots of high things around. Of course, no guarantees.