The elimination tree model for sparse unsymmetric matrices and an algorithm for constructing it have been recently proposed [Eisenstat and Liu, SIAM J. Matrix Anal. Appl., 26 (2005) and 29 (2008)]. The construction algorithm has a worst case time complexity ${\mathcal O}(mn)$ for an $n\times n$ unsymmetric matrix having $m$ nonzeros. We propose another algorithm that has a worst case time complexity of ${\mathcal O}(m\log n)$.