Let me clear up a misconception that is too OFTEN repeated.
THERE IS NO "BIG-TREE".
Or rather not in the sense that people mean it. The vast majority of people are connected to the tree of 37,039,751 people (and counting), sideways. That is, through marriage. In this sense merging ANY profile at all, even with another tree that is NOT yet connected to the "big-tree", is only one merge away from the "big-tree".
What I think people actually MEAN when saying "merge into the big tree", is merging into the main-lines of the historical profiles. By which one instantly gains "millions" of cousins. But then again, this too is ALSO just like any other merge. Because you could merge your 5th great-grandfather with somebody else who is also NOT connected, only to have a 3rd person merge that into the "main-lines".
So the ONLY way to have a private tree is NO merges EVER, and thus, never inviting ANY one to your tree. Because as soon as you invite a single person, they can invite someone else, and eventually someone will do that first merge.
Also you can't have it both ways. Inviting someone and then expecting them to not invite or merge anybody else, just doesn't work, and is totally unfair.