The elimination tree model for 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 is of time complexity ${\mathcal O}(mn)$ for an $n\times n$ unsymmetric matrix having $m$ nonzeros. We propose another algorithm with a time complexity of ${\mathcal O}(m\log n)$.