Convex combination of first and second eigenvalues of trees

For a graph $G$, let $\lambda_1(G)$ and $\lambda_2(G)$ denote the largest and the second largest adjacency eigenvalue of $G$. The sum $\lambda_1(G) + \lambda_2(G)$ is called the \emph{spectral sum} of $G$. We investigate the spectral sum of trees of order $n$ and determine the extremal trees that achieve maximum/minimum. Moreover, for any $\alpha \in [0,1]$, we determine the extremal trees which maximize the convex combination $\alpha \lambda_1 + (1-\alpha)\lambda_2$ in the class of $n$-vertex trees.