  Let $\mathcal{L}(T,\lambda)=\sum_{k=0}^n(-1)^{k}c_{k}(T)\lambda^{n-k}$ be the
characteristic polynomial of its Laplacian matrix of a tree $T$. This paper
studied some properties of the generating function of the coefficients sequence
$(c_0, \cdots, c_n)$ which are related with the matching polynomials of
division tree of $T$. These results, in turn, are used to characterize all
extremal trees having the minimum Laplacian coefficient generation function and
the minimum incidence energy of trees with described maximum degree,
respectively.
