Let $T=(V,E)$ be a finite tree with vertex set $V$ and set of edges $E$. Let $A\subset V$ be a subset containing all vertices of degrees $1$ and $2$. Assume that for every triple $(p,q,r)\in\binom{A}{3}$ of vertices in $A$ we know its middle point.
In the left picture $b$ is the middle point of $(p,q,r)$. In the right picture $q$ is the middle point of $(p,q,r)$.
Is this amount of information sufficient to uniquely determine $T$, or is there a second tree with the same vertex set, the same middle point information, but different set of edges?