For a weighted tree, the Lin--Lu--Yau Ricci curvature admits an explicit formula in terms of the edge weights. Consequently, the constant-curvature equation is equivalent to an eigenvalue problem for an edge-indexed Ricci matrix $R_T$. Building on the spectral characterization of discrete Einstein metrics on trees, we classify all finite trees whose discrete Einstein metric has positive curvature, equivalently all trees satisfying $λ_{\max}(R_T)<0$. For caterpillars with spine order $m\ge 12$, this occurs precisely for the endpoint families $T_m(a,0,\ldots,0,b)$ with $1\le a,b\le 3$ and $(a,b)\ne(3,3)$. The remaining cases $3\le m\le 11$ are settled by an exact finite verification using rational characteristic polynomials and Sturm root counts. We also determine the zero level set $λ_{\max}(R_T)=0$: among caterpillars, it consists of the stable family $(3,0,\ldots,0,3)$ together with nine exceptional short-spine caterpillars, while $S_3^2$ is the unique non-caterpillar zero example.