Summary: The Balaban index (also called \(J\) index) of a connected graph \(G\) is defined as \(J=J(G)=\frac{| E(G)| }{\mu+1}\sum_{uv\in E(G)}\big(\sqrt{\sigma_G(u)\sigma_G(v)}\big)^{-1/2}\), where \(\sigma_G(u)=\sum_{w\in V(G)}d_G(u,w)\) and \(\mu\) is the cyclomatic number. Balaban index has been used in various QSAR and QSPR studies. We characterize the trees with the maximum Balaban index among all the trees with \(n\) vertices and either the maximum degree \(\Delta\), or a given degree sequence, or \(k\) pendent vertices. In addition, the tree with the minimum Balaban index and \(n\) vertices and the maximum degree \(\Delta\), and the tree with the maximum Balaban index and \(n\) vertices and the maximum degree \(\Delta\) and \(k\) pendent vertices are also determined. On the other hand, we find that a lemma without proof in our paper [MATCH Commun. Math. Comput. Chem. 63, 799–812 (2010; Zbl 1265.05182)] is incorrect, and so are several related theorems. In the appendix, we rework the lemma and theorems, and give a new character for the graph with maximum Balaban index among graphs with \(n\) vertices. Several open problems on graphs with extreme Balaban indices are proposed.