Extremal problems for trees with given segment sequence

A segment of a tree T is a path whose end vertices have degree 1 or at least 3, while all internal vertices have degree 2. The lengths of all the segments of T form its segment sequence, in analogy to the degree sequence. We address a number of extremal problems for the class of all trees with a given segment sequence. In particular, we determine the extremal trees for the number of subtrees, the number of matchings and independent sets, the graph energy, and spectral moments.