Let $T_v(z)$ be the (exponential) generating function for vertex-rooted (non-plane) trees.
Im trying to construct the generating function $T_e(z)$ for edge-rooted trees from this.
I know the coefficients for these GFs are well known, I'm trying to verify that my idea is correct, since i'd like to use it on more specific cases of trees (bounded maximum degree and such).
So my idea was taking a designated edge, which is the same as 2 vertices, so that translates to $2z$ in the GF world. Then add a vertex-labelled tree on to each of the vertices, in the GF world that should give a factor $(T_v(z))^2$. So I get: $$T_e(z) = 2z(T_v(z))^2$$
Is this correct?
Thanks a lot