|Supplementary Appendix 1. Analysis of Fossil Taxa.
We performed an analysis of ancestral habitat distribution for the root of the phylogeny of actinopterygian fish that included many fossil taxa. For this analysis, we used the tree of higher-level actinopterygian relationships presented by Gardner et al. (2005). This tree is well-resolved, includes both living and fossil clades, and is consistent with the molecular phylogeny (for relationships among the living taxa). Other phylogenies could have been used, but did not meet these criteria. For example, the analysis of Friedman & Blom (2006) does not contain any living clades, making it difficult to relate the results to the tree of living taxa (but note that the tree of Gardner et al. 2005 contains many of the same fossil taxa used by Friedman & Blom 2006). The analysis by Hurley et al. (2007; their Fig. 2) contains extensive sampling of fossil taxa, but is largely unresolved near the root of actinopterygians, and the rooting used by those authors makes the tree somewhat difficult to reconcile with a tree of living taxa (e.g., the authors root the tree with Acipenser [Chondrostei] and Polypterus [Polypteriformes], which are then placed together to the exclusion of all fossil taxa, but the analysis by Gardner et al.  suggests that many of the included fossil taxa should be on branches separating these living clades).
We reconstructed habitat use on the phylogeny of Gardner et al. (2005) using Mesquite, version 2.73 (Maddison & Maddison 2010). We generally coded genera as marine or freshwater following the designations in Friedman & Blom (2006). However, we considered Cheirolepis to be primarily freshwater (but recognizing the possibility that some species may also range into estuarine environments). For genera not included by Friedman & Blom (2006), we searched the Paleobiology Database (http://paleodb.org/cgi-bin/bridge.pl?a=home) and the literature for additional information on habitat, including the following: Birgeria, Pteronisculus (Schaeffer & Magnus 1976), Australosomus (Stensiö 1932), and Perleidus (Lombardo et al. 2011). There did not appear to be sufficient information on the details of species distributions to create a third category for species that spanned both freshwater and saltwater. We coded the living clades Accipenseriformes (equivalent to Chondrostei in our Fig. 1) and Neopterygia (including all living actinopterygians exclusive of Polypteriformes and Chondrostei) as unknown for habitat, as both occur in both freshwater and saltwater. However, alternate codings for these taxa generally have little impact on the results (not shown). We performed both parsimony and likelihood analyses. The phylogeny of Gardner et al. (2005) does not include branch length information. Likelihood analyses assumed that all branch lengths were equal, and utilized the two-rate model (allowing for different rates of transition between freshwater and saltwater environments). The branch lengths could have been arbitrarily ultrametricized, but this would be inappropriate given that most of the included taxa are fossils and so should not be treated as having branches that are long and extend to the present day. In theory, the ages of fossil taxa could also have been used to estimate branch lengths, but even though the stratigraphic occurrence of fossil taxa is known, the actual branching times among lineages are not.
Results from parsimony and likelihood were generally similar (Supplementary Figure 2). Importantly, both methods show the ancestor of all living actinopterygians as having occurred in freshwater, consistent with the reconstructions from living taxa. Both methods also show the most recent common of Chondrostei and Neopterygia (including all other living actinopterygians, exclusive Chondrostei and Polypteriformes) as having been marine. The analyses of living taxa show this branch as ambiguous. The fossil analyses suggest that additional freshwater lineages may have marine ancestors than are indicated by the analyses of living taxa alone. However, such a pattern further supports the importance of marine extinctions in driving patterns of richness in extant actinopterygians.
We acknowledge that more extensive analyses of the phylogeny and habitat of fossil actinopterygians would be highly desirable. However, resolving the phylogeny of fossil and living actinopterygians is somewhat beyond the scope of the present study, which is focused on richness patterns in living fish.
References (not in main text)
Gardiner, B. G., Schaeffer, B. & Masserie, J. A. 2005 A review of the lower actinopterygian phylogeny. Zool. J. Linn. Soc. 144, 511–525.
Hurley, I. A., Lockeridge-Mueller, R., Dunn, K. A., Schmidt, E. J., Friedman, M., Ho, R. K., Prince, V. E., Yang, Z., Thomas, M. G. & Coates, M. I. 2007 A new time-scale for ray-finned fish evolution. Proc. R. Soc. Lond. B. 274, 489–498.
Lombardo, C., Sun, Z. Y.. Tintori, A., Jiang, D. Y. & Hao, W. C. 2011 A new species of the genus Perleidus (Actinopterygii: Perleidiformes) from the Middle Triassic of Southern China. Bollettino della Società Paleontologica Italiana 50, 75–83.
Schaeffer, B. & Mangus, M. 1976 An early Triassic fish assemblage from British Columbia. Bull. Am. Mus. Nat. Hist. 5, 519–563.
Stensiö, E. 1932 Triassic Fishes from East Greenland collected by the Danish expeditions in 1929-1931. Meddelelser om Grønland 83, 1-305.