  The \emph{Wiener polynomial} of a connected graph $G$ is the polynomial
$W(G;x) = \sum_{i=1}^{D(G)} d_i(G)x^i$ where $D(G)$ is the diameter of $G$, and
$d_i(G)$ is the number of pairs of vertices at distance $i$ from each other. We
examine the roots of Wiener polynomials of trees. We prove that the collection
of real Wiener roots of trees is dense in $(-\infty, 0]$, and the collection of
complex Wiener roots of trees is dense in $\mathbb C$. We also prove that the
maximum modulus among all Wiener roots of trees of order $n \ge 31$ is between
$2n-15$ and $2n-16$, and we determine the unique tree that achieves the maximum
for $n \ge 31$. Finally, we find trees of arbitrarily large diameter whose
Wiener roots are all real.
