It's a really similar generating function equation, actually T box to Z is ZT of Z. if you plug in ZT of Z of that equation, and divide by Z, you get the same equation as before. well, this is a proof that the number of binary trees within external nodes is the same as the number of binary trees within minus one internal nodes. there's easier proofs of that, but this is showing the consistency of the analysis and the ease of developing a commentorial construction to get a generating function in equation. 