Joyal's proof of Cayley's formula for the enumeration of trees

That formula of Cayley asserts that there are $n^{n-2}$ trees with vertices labeled $1,\dots,n$. This is a chapter in graph theory, in which graphs are “simple”. More precisely, let's call a graph structure on a set $V$ (called “vertices”) is the datum of a subset $E$ of 2-element subsets of~$V$, called “edges”. So an edge is of the form $\{a,b\}$, where $a$ and $b$ are distinct elements of $V$,…