§ Number of vertices in a rooted tree

Make sure the edges of the tree are ordered to point away from the root rr. So, for all edges (u,v)E(u, v) \in E, make sure that d(r,v)=d(r,u)+1d(r, v) = d(r, u) + 1. Create a function terminalterminal which maps every outward arc to its target. terminal:EVterminal: E \rightarrow V, terminal((u,v))=vterminal((u, v)) = v. This map gives us an almost bijection from edges to all vertices other than the root. So we have that E+1=V|E| + 1 = |V|. Each of the edges cover one non-root vertex, and we then +1+1 to count the root node. I found this much more intuitive than the inductive argument. I feel like I should attempt to "parallelize" inductive arguments so you can see the entire counting "at once".