There are two sorts of connected.
1. 'connected' with a small 'c' is when a profile is connected to parents, spouse, siblings, or children.
2. 'Connected' with a capital 'C' is when a profile is connected to the main tree (or 'big tree' as you called it). When they are related to or connected via marriages, parental or spousal relationships, sibling relationships and so on. You can tell if a person is Connected to the main tree if, at the bottom of their profile, you see the section which goes something like "Mary is 25 degrees from Emeril Lagasse, 24 degrees from Nigella Lawson"...and so on. The main tree is so huge, you could not see all of it in one go. From the Home Page: "Our shared tree has 37,965,304 profiles" - and some of them are going to be unconnected (so far).