On extremal trees with respect to the $F$-index

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.