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The eccentricity of a vertex $u$ in a connected graph $G$ is the distancebetween $u$ and a vertex farthest from it; the eccentric sequence of $G$ is thenondecreasing sequence of the eccentricities of $G$. In this paper, wedetermine the unique tree that minimises the Wiener index, i.e. the sum ofdistances between all unordered vertex pairs, among all trees with a giveneccentric sequence. We show that the same tree maximises the number of subtreesamong all trees with a given eccentric sequence, thus providing another exampleof negative correlation between the number of subtrees and the Wiener index oftrees. Furthermore, we provide formulas for the corresponding extreme values ofthese two invariants in terms of the eccentric sequence. As a corollary to ourresults, we determine the unique tree that minimises the edge Wiener index, thevertex-edge Wiener index, the Schulz index (or degree distance), and the Gutmanindex among all trees with a given eccentric sequence.

Wiener index, number of subtrees, and tree eccentric sequence