On asymptotic dimension of groups acting on trees

We prove the following theorem: Let $\pi$ be the fundamental group of a finite graph of groups with finitely generated vertex groups $G_v$ having asdim $G_v\le n$ for all vertices $v$. Then asdim$\pi\le n+1$. This gives the best possible estimate for the asymptotic dimension of an HNN extension and the amalgamated product.