A tree as a finite nonempty set with a binary operation

A (finite) acyclic connected graph is called a tree. Let $W$ be a finite nonempty set, and let $ H(W)$ be the set of all trees $T$ with the property that $W$ is the vertex set of $T$. We will find a one-to-one correspondence between $ H(W)$ and the set of all binary operations on $W$ which satisfy a certain set of three axioms (stated in this note).