  Let $G=(V,E)$ be a simple connected graph with vertex set $V(G)$ and edge set
$E(G)$. The third atom-bond connectivity index, $ABC_3$ index, of $G$ is
defined as $ABC_3(G)=\sum\limits_{uv\in
E(G)}\sqrt{\frac{e(u)+e(v)-2}{e(u)e(v)}}$, where eccentricity $e(u)$ is the
largest distance between $u$ and any other vertex of $G$, namely
$e(u)=\max\{d(u,v)|v\in V(G)\}$. This work determines the maximal $ABC_3$ index
of unicyclic graphs with any given girth and trees with any given diameter, and
characterizes the corresponding graphs.
