I wanted to quickly write this proof up, complete with pictures, so that I won't forget it again. In this post I'll give a combinatorial proof (due to Joyal) of the following: Theorem 1 (Cayley's Formula) The number of trees on $latex {n}&fg=000000$ labelled vertices is $latex {n^{n-2}}&fg=000000$. Proof: We are going to construct a… Continue reading Joyal’s Proof of Cayley’s Tree Formula