Characterizing trees where every diametral path shares an edge In a graph $G$, a _diametral path_ is a path of length $\text{diam}(G)$ joining two vertices that are at a distance $\text{diam}(G)$ from each other. Given a tree $T$, consider the set of all diametral paths of $T$. For example, if $T$ is say the star graph $S_4$, it is not true that the set of all diametral paths would share an edge. But when do they? How can we characterize trees where every diametral path has an edge in common? Looking at some examples, it seems always true for at least bicentered trees. The intuition would be that every such path passes through the center, and thus have the edge between the two centers in common.

Being bicentered is sufficient, but not necessary (consider $P_3$ with two edges attached at each end).

For odd diameter, being bicentered is necessary and sufficient. For even diameter $2r$, the condition would be that the (unique) center has exactly two neighbours leading to a vertex at distance $r$ from the center. This is not much of a characterization, but I doubt if you can find anything better.