Talk:Cayley's formula

$\phi$

Hello everybody,

can someone explain the meaning of $\phi$ in this proof? Thank you!

It's meant to be $\emptyset$. Fixed. Algebraist 14:34, 27 March 2008 (UTC)

why not Pruffer code proof

I think the proof based on Prueffer code is much clearer. —Preceding unsigned comment added by 141.211.62.162 (talk) 17:18, 9 January 2009 (UTC)

Proof

While I share David Eppstein's opinion that the proof of the Cayley formula given in the article is unnecessary tedious, I think there is a good case to be made for including one such proof. The proof in Prüfer sequence is barely readable, a pure punishment to the reader unfamiliar with the pseudo code. Alternatively, matrix tree theorem lists only a determinant without explicit calculation - that might also prove useful. Finally, a two line proof using the Lagrange inversion theorem would also be very helpful. My point is: this article needs a lot of work. Mhym (talk) 21:23, 13 April 2010 (UTC)

I kind of like the part in matrix tree theorem just above the determinant where it explicitly describes the eigenvectors and eigenvalues, shows that the eigenvectors it describes form a complete set, and observes that the product of the corresponding eigenvalues is exactly what we want. It's short (if you already believe the matrix tree theorem and know about determinants being the product of eigenvalues) and nicely relates Cayley's formula to its important generalization. But the description there is probably too short to be convincing as a proof. Aigner and Ziegler give four proofs (one already in Double counting (proof technique)#Counting trees), so that's at least an indication of notability for some of its proofs, but none of these seems particularly shorter than the others. They include the determinant calculation (their proof 2) but most of their effort in that one is on the proof of the matrix tree theorem itself; they just assert that the matrix in question has the right determinant by an "easy computation". —David Eppstein (talk) 22:01, 13 April 2010 (UTC)