"n nodes, there are n − 1 edges. (Think about it!) 4 There appears to be no consensus as to which of these concepts is the most basic. Some authors refer to a free tree simply as a “tree” — as though this were the “normal” kind of tree — and use the term rooted tree for the other kind. Other authors do the opposite. To avoid confusion, I’ll try to always use the full term (although I admit I’m one who considers rooted trees to be the more important, default concept).\n\nSo basically, if your goal is connecting all the nodes, and you have a free tree, you’re all set. Adding anything is redundant, and taking away anything breaks it. If this reminds you of Prim’s algorithm, it should. Prim’s algorithm produced exactly this: a free tree connecting all the nodes — and specifically the free tree with shortest possible total length. Go back and look at the final frame of Figure 5.14 and convince yourself that the darkened edges form a free tree. For this reason, the algorithm is often called Prim’s minimal spanning tree algorithm. A “spanning tree” just means “a free tree that spans (connects) all the graph’s nodes.” Keep in mind that there are many free trees one can make with the same set of vertices. For instance, if you remove the edge from A to F, and add one from anything else to F, you have a different free tree."