Let $G$ be a finite simple graph with Laplacian polynomial $\psi(G,\lambda)$ $=$ $\sum_{k=0}^{n}(-1)^{n-k}c_k(G)\lambda^k$. In an earlier paper, we computed the coefficient of $c_{n-4}$ for trees with respect to some degree-based graph invariant. The aim of this paper is to continue this work by giving an exact formula for the coefficient $c_{n-5}$ in the polynomial $\psi(G,\lambda)$. As a consequence of this work, the Laplacian coefficients $c_{n-k}$, $k = 2, 3, 4, 5$, for some know trees were computed.