= Solution
Let $f:X\to T$ be a $(\lambda,\varepsilon)$-<quasi-isometry> to a tree. The image under $f$ of every geodesic segment in $X$ is a $(\lambda,\varepsilon)$-quasigeodesic in $T$. By the <Morse lemma for quasi-geodesics>, it lies within a constant $R=R(\lambda,\varepsilon)$ of the tree geodesic with the same endpoints.
Consider a geodesic triangle in $X$. A point $p$ on one side maps within $R$ of the corresponding side of the comparison triangle in $T$. Every geodesic triangle in a tree is $0$-thin, so that comparison side is contained in the other two sides. Those two tree sides are in turn within $R$ of the images of the other two sides of the original triangle. Hence some point $q$ on one of those sides satisfies
$$
d_T(f(p),f(q))\leq2R.
$$
The lower quasi-isometry inequality gives
$$
d_X(p,q)\leq\lambda(2R+\varepsilon).
$$
Thus $X$ is <Gromov-hyperbolic metric space> with $\delta=\lambda(2R+\varepsilon)$.
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