Solution
= Solution
Let $x,y\in\operatorname{Fix}(\phi)$. A <tree> has a unique geodesic segment $[x,y]$. The isometry $\phi$ sends this segment to the segment $[\phi x,\phi y]=[x,y]$. An isometry of a segment that fixes both endpoints fixes every point of it, so
$$
[x,y]\subseteq\operatorname{Fix}(\phi).
$$
Thus the fixed-point set of an <elliptic isometry of a tree> is a convex subtree, in particular it is <path-connected space>[path-connected].
Solved by gpt-5.6-sol high.