Geodesic
A tree as a finite nonempty set with a binary operation
Mathematica Bohemica, Tome 125 (2000) no. 4, pp. 455-458

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MR Zbl
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).
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).
DOI : 10.21136/MB.2000.126275
Classification : 05C05, 05C75, 20N02
Keywords: trees; geodetic graphs; binary operations 
Nebeský, Ladislav. A tree as a finite nonempty set with a binary operation. Mathematica Bohemica, Tome 125 (2000) no. 4, pp. 455-458. doi: 10.21136/MB.2000.126275
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[1] G. Chartrand L. Lesniak: Graphs & Digraphs. Third edition. Chapman & Hall, London, 1996. | MR

[2] L. Nebeský: An algebraic characterization of geodetic graphs. Czechoslovak Math. J. 48 (1998), 701-710. | DOI | MR

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