  We prove the existence of an upper bound on the asymptotic dimension of tree
amalgamations of locally finite quasi-transitive connected graphs. This
generalises a result of Dranishnikov for free products with amalgamation and a
result of Tselekidis for HNN-extensions of groups to tree amalgamations of
graphs. As a corollary, we obtain an upper bound on the asymptotic dimension of
a multi-ended quasi-transitive locally finite graph based on any of their
factorisations.
