I think perhaps we have a situation a bit analogous to the optical illusion (hopefully everyone reading this is familiar with), that switches between looking like a vase and looking like two faces looking at each other -
The analogy being that the meaning of "Tree" flips back and forth, just as what one sees in that situation flips back and forth.
At least, it seems to do so for Geni, which uses it both ways on an inconsistent basis - Geni officially defined "Tree" one way but then used it the other way in their term "World Tree" - thus undercutting their own attempts to distinguish between the terms 'Tree' and 'Forest'.
It definitely is correct to say there are many, many Trees on Geni.
The way you (Erica) think of it also makes sense.
[they use two different definitions of Tree]
Even using the more limited meaning of the term 'Tree' - with many, many Trees on Geni making up Forests/the Forest - each Tree still does not belong to just one person.