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.