For every tree X and point p in it, if another tree Y has a homeomorphic space of connected subcontinua containing a point q, then X and Y are homeomorphic by a map sending p to q.
Illanes, Finite graphs have unique hyperspaces Cn(X), Topology Proc., 27 (2003), 179–188
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Uniqueness of the hyperspaces $C(p,X)$ in the class of trees
For every tree X and point p in it, if another tree Y has a homeomorphic space of connected subcontinua containing a point q, then X and Y are homeomorphic by a map sending p to q.