We correct the proof of the Brundobler-Elser formula (BEF) provided in Volkov and Ostrovsky (2004 J. Phys. B: At. Mol. Opt. Phys. 37 4069) and continued in the appendix of Volkov and Ostrovsky (2005 J. Phys. B: At. Mol. Opt. Phys. 38 907). After showing that some changes of variables employed in these articles are used erroneously, we propose an alternative change of variables which solves the problem. In our proof, we reveal the connection between the BEF for a general N-level Landau-Zener system and the exactly solvable bow-tie model. The special importance of the diabatic levels with maximum/minimum slope is emphasized throughout.