Summary: The independence polynomial $ω(G)$ of a graph $G$ is a polynomial whose $k$-th coefficient is the number of selections of $k$ independent vertices in $G$. The main result of the paper is the identity $$ω(T- u)ω(T-v)-ω(T)ω(T-u-v)=-(-x)\sp{d(u,v)}ω(T-P)ω(T-[P])$$ where $u$ and $v$ are distinct vertices of a tree $T$, $d(u,v)$ is the distance between them and $P$ is the path connecting them; the subgraphs $T-P$ and $T-[P]$ are obtained by deleting from $T$ the vertices of $P$ and the vertices of $P$ together with their first neighbors. A conjecture of R. E. Merrifield and H. E. Simmons is proved with the help of this identity, which is also compared to some previously known analogous results.