Ore (1962) noted that not only does a tree possesses a unique shortest path between any two vertices, but that there also exist also other connected
graphs having the same property. He termed all such graphs "geodetic graphs"
and asked for a characterization of such graphs.

The numbers of geodetic graphs on , 2, ... nodes are 1, 1, 2, 4, 10, 23, 66, 185, 586, 1880,
6360, ... (OEIS A337179).

Frasser, C. E. "-Geodetic Graphs and Their Application to the Topological Design
of Computer Networks." In Proc. Argentinian Workshop on Theoretical Computer
Science, 28 JAIIO-WAIT'99. pp. 187-203, 1999.Gorovoy, D. and
Zmiaikou, D. "On Graphs with Unique Geoodesics and Antipodes." 19 Nov 2021.
https://arxiv.org/abs/2111.09987.Ore,
O. Theory of Graphs. Providence, RI: Amer. Math. Soc., 1962.Parthasarathy,
K. R. and Srinivasan, N. "Some General Constructions of Geodetic Blocks."
J. Combin. Th.33, 121-136, 1982.Sloane, N. J. A.
Sequence A337179 in "The On-Line Encyclopedia
of Integer Sequences."Stemple, J. G.; and Watkins, M. E.
"On Planar Geodetic Graphs." J. Combin. Th.4, 101-117, 1968.