For a graph G with domination number γ, Hedetniemi, Jacobs and Trevisan (2016) proved that m G [ 0 , 1 ) ≤ γ , where m G [ 0 , 1 ) means the number of Laplacian eigenvalues of G in the interval [ 0 , 1 ) . Let T be a tree with diameter d. In this paper, we show that m T [ 0 , 1 ) ≥ ( d + 1 ) / 3 . All trees achieving the lower bound are completely characterized. Moreover, we prove that the domination number of a tree is ( d + 1 ) / 3 if and only if it has exactly ( d + 1 ) / 3 Laplacian eigenvalues less than one. As an application, it also provides a new type of tree, which shows the sharpness of the inequality due to Hedetniemi, Jacobs and Trevisan.