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Appendix S1. Stochastic approaches: discrete intervals and times of observation

Fig. S1. An example of a stochastic approach in time-scaling is generating continuous-time first and last appearance dates from discrete intervals, as shown in (a), by randomly sampling from dates within those intervals under a uniform probability distribution. (b) This produces a set of continuous-time dates within the appropriate intervals. (c) This sampling can be repeated many times to produce many potential sets of appearance dates.

Fig. S2. Time-scaled phylogenies describe the amount of shared and unshared evolutionary history among instantaneously sampled populations, not persistent lineages. The chosen times of observation for a set of morphotaxa can produce very different realizations of phylogeny.

Fig. S3. The calculated expected probability of sampling extinct clades of unknown size predicts the proportion of clades from simulations which were either sampled or still extant after 1000 time-units.

Fig. S4. Gamma models which consider the potential for unobserved extinct lineages provide a better fit to the simulated distributions of true amount of total inferable unobserved evolutionary history.

Fig. S5. QQ plot of candidate probability models against simulated Δ values generated under budding cladogenesis with branching, extinction and sampling rates of 0·1 per Ltu.

Fig. S6. QQ plot of candidate probability models against simulated Δ values generated under budding cladogenesis with branching and extinction rates of 0·5 per Ltu and a sampling rate of 0·1 per Ltu.

Fig. S7. QQ plot of candidate probability models against simulated Δ values generated under bifurcating cladogenesis with branching, extinction and sampling rates of 0·1 per Ltu.

Fig. S8. QQ plot of candidate probability models against simulated Δ values generated under budding cladogenesis with branching and extinction rates of 0·1 per Ltu and sampling rates of 0·5 per Ltu.

Fig. S9. The per-run median of the median squared error of node ages reveals that (a) under budding, the basic method has less error in node ages but (b) under the terminal-taxon model has similar amounts of error as the cal3 method.

Fig. S10. cal3 correctly resolves slightly collapsed clades on secondarily degraded trees than randomly resolving nodes (‘Rand-Res’), but this increase is minor compared to the total number of clades which can be correctly inferred by resolving absolutely with respect to temporal order (‘Time-Res’).

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