  In the paper [I. Gutman, N. Trinajsti\'c, Chem. Phys. Lett. 17 (1972), 535],
it was shown that total $\pi$-electron energy ($E$) of a molecule $M$ depends
on the quantity $\sum_{v\in V(G)}d_{v}^{2}$ (nowadays known as the "first
Zagreb index"), where $G$ is the graph corresponding to $M$, $V(G)$ is the
vertex set of $G$ and $d_{v}$ is degree of the vertex $v$. In the same paper,
the graph invariant $\sum_{v\in V(G)}d_{v}\tau_{v}$ (where $\tau_{v}$ is the
connection number of $v$, that is the number of vertices at distance 2 from
$v$) was also proved to influence $E$, but this invariant was never restudied
explicitly. We call it "modified first Zagreb connection index" and denote it
by $ZC_{1}^{*}$. In this paper, we characterize the extremal elements with
respect to the graph invariant $ZC_{1}^{*}$ among the collection of all
$n$-vertex chemical trees.
