The reverse Wiener index of a connected graph $G$ is defined as $$\Lambda \left(G\right)=\frac{1}{2}n(n-1)d-W\left(G\right),$$ where $n$ is the number of vertices, $d$ is the diameter, and $W\left(G\right)$ is the Wiener index (the sum of distances between all unordered pairs of vertices) of $G$. We determine the $n$-vertex non-starlike trees with the first four largest reverse Wiener indices for $n\ge 8$, and the $n$-vertex non-starlike non-caterpillar trees with the first four largest reverse Wiener indices for $n\ge 10$.