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).
Examples of geodetic graphs include barbell graphs, block graphs, cacti graph
lacking even cycles (Gorovoy and Zmiaikou 2021), complete
graphs 
(Gorovoy and Zmiaikou 2021), odd cycle graphs 
(Gorovoy and Zmiaikou 2021),
lollipop graphs, trees
(Gorovoy and Zmiaikou 2021), triangular snake
graphs, and windmill graphs.
-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.