3E. Different Distances

This tree is a simple path $1-2-3-4-5$. For example, $S=\{2,3\}$ is good, because for any two distinct vertices $u \ne v$, at least one of $x=2$ or $x=3$ has different distances to them, so $dist(u,x) \ne dist(v,x)$. On the other hand, $S=\{3\}$ is not good, because vertices $2$ and $4$ have the same distance to $3$, so $dist(2,3)=dist(4,3)=1$, and there is no other vertex in $S$ to distinguish them.

This tree has a branching at vertex $2$ (edges $1-2$, $2-3$, $2-4$). For example, $S=\{1,3\}$ is good, since every pair of vertices can be distinguished using either $x=1$ or $x=3$. However, $S=\{2\}$ is not good, because vertices $1$ and $3$ have the same distance to $2$, so $dist(1,2)=dist(3,2)=1$.