a–c, Diversification analysis of a timetree (about 114,000 tips) simulated from a birth–death process that exhibits a mass extinction event at around 5 Myr before present. a, LTT of the generated tree (long-dashed curve), dLTT of the true model that generated the tree (continuous curve) and dLTT of a maximum-likelihood fitted model (short-dashed curve) are shown. The fitted dLTT is practically identical to the true dLTT and thus is covered by the latter. b, True speciation and extinction rates (continuous curves), compared to fitted speciation and extinction rates (dashed curves). There is considerable disagreement between the fitted and true λ and μ, despite the fact that the allowed model set could—in principle—approximate the true rates reasonably well. c, Pulled diversification rate (PDR) of the true model (continuous curve), compared to the pulled diversification rate of the fitted model (dashed curve). d–f, Diversification analysis of a timetree (about 785,000 tips) simulated from a birth–death process that exhibits a rapid radiation event at around 5 Myr before present and a mass extinction event at around 2 Myr before present. d–f are analogous to a–c. There is considerable disagreement between the fitted and true λ and μ, despite the fact that the allowed model set could—in principle—approximate the true rates reasonably well. Extended Data Figure 7 provides the fitting results when μ is fixed to its true value. g–i, Diversification analyses of an extant timetree of 79,874 seed plant species, performed either by fitting λ and μ on a grid of discrete time points or by fitting the parameters of generic polynomial or exponential functions for λ and μ. g, LTT of the tree, dLTT of the grid-fitted model and dLTT of the fitted parametric model. h, Speciation and extinction rates predicted by the grid-fitted model or the fitted parametric model. i, Pulled diversification rate predicted by the grid-fitted model and the fitted parametric model. Further details are provided in Supplementary Information sections S.10 and S.11.