Lexicographical ordering by spectral moments of trees with a given bipartition

Lexicographic ordering by spectral moments ($S$-order) among all trees is discussed in this‎ ‎paper‎. ‎For two given positive integers $p$ and $q$ with $p\leqslant q$‎, ‎we denote $\mathscr{T}_n^{p‎, ‎q}=\{T‎: ‎T$ is a tree of order $n$ with a $(p‎, ‎q)$-bipartition\}‎. Furthermore‎, ‎the last four trees‎, ‎in the $S$-order‎, ‎among $\mathscr{T}_n^{p‎, ‎q}\,(4\leqslant p\leqslant q)$ are characterized‎.