Laplacian energy of diameter 3 trees

Let TnTn be the set of all trees of diameter 3 and nn vertices. We show that the Laplacian energy of any tree in TnTn is strictly between the Laplacian energy of the path PnPn and the star SnSn, partially proving the conjecture that this holds for any tree. We also give a total order by the Laplacian energy in TnTn. Moreover, we show that this order depends only on the algebraic connectivity of the tree: the Laplacian energy increases as the algebraic connectivity decreases in TnTn.