  We consider the following problem that we call the Shortest Two Disjoint
Paths problem: given an undirected graph $G=(V,E)$ with edge weights $w:E
\rightarrow \mathbb{R}$, two terminals $s$ and $t$ in $G$, find two internally
vertex-disjoint paths between $s$ and $t$ with minimum total weight. As shown
recently by Schlotter and Seb\H{o} (2022), this problem becomes NP-hard if
edges can have negative weights, even if the weight function is conservative,
there are no cycles in $G$ with negative total weight. We propose a
polynomial-time algorithm that solves the Shortest Two Disjoint Paths problem
for conservative weights in the case when the negative-weight edges form a
constant number of trees in $G$.
