  In a study on the structure--dependency of the total $\pi$-electron energy
from 1972, Trinajsti\'c and one of the present authors have shown that it
depends on the sums $\sum_{v\in V}d(v)^2$ and $\sum_{v\in V}d(v)^3$, where
$d(v)$ is the degree of a vertex $v$ of the underling molecular graph $G$. The
first sum was later named {\it first Zagreb index} and over the years became
one of the most investigated graph--based molecular structure descriptors. On
the other hand, the second sum, except in very few works on the general first
Zagreb index and the zeroth--order general Randi\'c index, has been almost
completely neglected. Recently, this second sum was named {\em forgotten
index}, or shortly the $F$-{\em index}, and shown to have an exceptional
applicative potential. In this paper we examine the trees extremal with respect
to the $F$-index.
