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Let Tn(d) be the set of all trees with n vertices, diameter d and perfect matchings.We will show that the Laplacian energy of any tree in Tn(d), where d =4, 5,is no less than the Laplacian energy of the path Pn.Recently, it has shown that theLaplacian energy of any tree with n vertices is no more than the Laplacian energy of thestar Sn.Thus, it partly shows that the conjecture given by Radenkovi(c) and Gutman is true.