Contractible independence complexes of trees

We show that the independence complex of a tree is contractible if and only if it can be reduced to a path $P_n$ with $n\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}3)$ by a sequence of truncation moves at branching points. As a consequence of our method, we also characterize the trees for which the independence polynomial evaluated at $-1$ is equal to $1$ or $-1$.