List homomorphism problems for signed trees

We consider homomorphisms of signed graphs from a computational perspective. In particular, we study the list homomorphism problem seeking an input graph $(G,\sigma)$, equipped with lists $L(v) \subseteq V(H), v \in V(G)$, allowed images, to fixed target $(H,\pi)$. The complexity similar without (corresponding all being $L(v)=V(H)$) has been previously classified by Brewster and Siggers, but version remains open appears difficult. illustrate this difficulty classifying when $H$ is tree (with possible loops). tools develop will be useful for classifications other classes graphs, in future companion paper using them classify certain irreflexive graphs. structure trees polynomial cases interesting, suggesting that class general which problems are may have nice structure, analogous so-called bi-arc (which characterized unsigned graphs).