Note on the hybridization number and subtree distance in phylogenetics

For two rooted phylogenetic trees TT and T′T′, the rooted subtree prune and regraft distance between TT and T′T′ has often been used as a replacement for the hybridization number of TT and T′T′. However, Baroni et al. [M. Baroni, S. Grünewald, V. Moulton, C. Semple, Bounding the number of hybridisation events for a consistent evolutionary history, J. Math. Biol. 51 (2005) 171–182] constructed particular instances that showed that both the difference and the ratio between this number and the distance can be arbitrarily large. In this note, we show that the difference and ratio values obtained in the above reference of Baroni et al. are the best possible, thus answering a problem posed in [C. Semple, Hybridization networks, in: O. Gascuel, M. Steel (Eds.), Reconstructing Evolution: New Mathematical and Computational Advances, Oxford University Press, 2007, pp. 277–314].