§ Number of vertices in a rooted tree
Make sure the edges of the tree are ordered to point away from the root .
So, for all edges , make sure that .
Create a function which maps every outward arc
to its target. , .
This map gives us an almost bijection from edges to all vertices other than
the root. So we have that . Each of the edges cover one non-root
vertex, and we then 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".