Last Christmas, we were on the Wonder so all of this reflects that. I really only remember the one really large tree. There may have been others I'm just not remembering, though. The tree is right in the main lobby on the third floor, but extends up well past the railing of the fourth floor. We in fact took several pictures on the 4th floor right nearby; it's a little closer to a "regular" tree size from there!
To answer your question (which is not that weird - everyone has different aspects of holidays that are important to them), I think it might depend on what you mean by "near", and what you mean by "later in the night". Earlier in the night - probably as long as there are any scheduled activities, the lobby will range from busy to occasional folks and it might be awkward to do much of anything on the third floor (the photographers will be out for quite a while, even, and you can't go near the tree then except for photos). But, there've been times I've found myself late at night (well after midnight) on the ship with no one around, other than the occasional cast member (usually doing some cleaning or maintenance). I imagine if you were there really late, no one other than the cast members might even know you were there!
But, there's a better option, I think. Up on the fourth floor, near the top of the tree, there were some of the lounge seats/booths. These were right near the tree. Last Christmas, they were often blocked off for the socially distanced characters, but I expect they will be more available this time around. I don't think it would be all that odd or awkward to be at one of the seating places in that area opening gifts (I mean, I guess if you are opening giant bags of gifts it might be, but I'm expecting you mean something less!). It would still be near the tree, and if it's after the larger crowds are gone, would probably be more private-ish, as well.
Of course, others could have the same idea, or the arrangement could be different or whatever, but I think that might work?
