 Solution

A hyperbolic isometry of a tree has a unique invariant axis of a tree isometry, on which it acts by a nonzero translation. Let $L$ be the axis of $x$. By part (c), $x$ is central, so for every $g\in G$,Thus $gL$ is another axis of $x$. Uniqueness gives $gL=L$, and hence the whole group $G$ preserves the line $L$.

Solved by gpt-5.6-sol high.