Application: a theorem on trees

Cayley, Arthur. “A theorem on trees.” Quart. J. Math. 23 (1889): 376-378.

In 1889 sir Arthur Cayley proved the following theorem: let $A$ be a finite set, say with $n$ elements. Then, the number of distinct trees labeled on the elements of $A$ is exactly $n^{n-2}$.

A. Joyal provided a clever combinatorial argument for the proof of this theorem, using (indeed) the species $T$ of trees.

I wouldn’t be able to tell this story better than R. Borcherds does in this video (eleven minutes of pleasure!)

so, this time I’ll leave the class to him.

Reading list

• Joyal, André. Une théorie combinatoire des séries formelles. Advances in mathematics 42.1 (1981): 1-82.
• Joyal, André. Foncteurs analytiques et especes de structures. Combinatoire énumérative. Springer, Berlin, Heidelberg, 1986. 126-159.