Most known T. rex crania containing teeth were examined (Supplementary Data 1, available online at www.vertpaleo.org/ jvp/JVPcontents.html). Tarbosaurus Maleev, 1955, was excluded as there is no consensus on its taxonomy and it appears to be a distinct species (Holtz, 2001; Hurum and Sabath, 2003; Carr and Williamson, 2004). Data from Dilophosaurus Welles, 1970; Liliensternus Welles, 1984; Ceratosaurus dentisulcatus Madsen and Welles, 2000 (?= Ceratosaurus nasicornis Marsh, 1884); Masiakasaurus Sampson et al. 2001; ‘ Indosuchus ’; Majungatholus Sues and Taquet, 1979; Baryonyx Charig and Milner, 1986; Suchomimus Sereno et al., 1998; Allosaurus Marsh 1877; Acrocanthosaurus Stovall and Langston, 1950; Carcharodontosaurus Stromer, 1931; Gorgosaurus Lambe, 1914; Daspletosaurus Russell, 1970; Tyrannosaurus rex; Troodon Leidy, 1856; Saurornithoides junior Barsbold, 1974; Bambiraptor Burnham et al., 2000; Deinonychus Ostrom, 1969 a; Dromaeosaurus Matthew and Brown, 1922; and Velociraptor Osborn, 1924 were used to provide context for teeth of Tyrannosaurus rex within theropod dental morphospace (see Smith et al., 2005, for data). Growth-related change is important in paleobiology (e.g., Currie, 2003b). However, as ontogeny in Tyrannosaurus rex is currently poorly understood, data were excluded from problematic specimens (see Molnar and Carpenter, 1989; Carr, 1999; Brochu, 2002; Carr and Williamson, 2004), such as LACM 28741 (‘ Aublysodon ’) and CMNH 7541 (‘ Nanotyrannus ’), to be dealt with separately. These specimens are likely juveniles of T. rex (see Carr, 1999; Holtz, 2001), and have been synonymized as such (Carr and Williamson, 2004). However, a consensus is lacking (see Currie, 2003a; Currie et al., 2003), and there are new data (J. Peterson, pers. comm., 2002) that merit consideration. I am not comfortable coding teeth as pertaining to T. rex unless there is general agreement that they are such, nor is it wise to use data from a problematic specimen to support or refute a T. rex affinity for that individual. Therefore, I included only those data that are currently unquestioned as pertaining to T. rex. As tyrannosaurid taxonomy becomes clearer, the dentitions of problematic specimens should be compared against those of well-supported taxa. In the absence of this and in the absence of data for proven juvenile teeth for T. rex, discussions of ontogenetic change in its dentition are premature (e.g., Senter and Robins, 2003). Measurements and Counts Measurements were made directly with electronic calipers or on digital images using SigmaScan®. Denticles were counted with a Hensoldt-Wetzlar 8X lens possessing a mm-scale reticle. Data cases are averages of five replicate measurements (measurement repeatability was assessed by Smith, 2002). Data were included from teeth regarded as being fully erupted (teeth that are erupting, are reconstructed, or are not accessible because the specimen is on display were omitted). Orientation terminology (Fig. 1A) follows Smith and Dodson (2003). Studies of tyrannosaurid dentitions must pay particular attention to the premaxillary and first dentary teeth. The basal short axes of these crowns are mesiodistal rather than labiolingual in orientation (as in most theropods), and the long axes (sensu the basoapical axis in human incisors, see Minagi et al., 1999) are oriented labiolingually rather than mesiodistally (Fig. 1B, C). The variables used in this article (Fig. 1D, E) were derived or discussed in detail by Smith et al. (2005); they are noted briefly here. Size was described using crown base length (CBL), crown base width (CBW), crown height (CH), and apical length (AL). CBL and CBW were measured in a horizontal plane referenced approximately to point B of Smith et al. (2005). Basal shape was described using the crown base ratio (CBR = CBW/CBL); crown “squatness” was assessed using the crown height ratio (CHR = CH/CBL). Apex displacement from the crown base center and crown curvature were described using the crown angle (CA). Crown angle values were calculated using the Law of Cosines and solving for: where a = CBL, b = AL, and c = CH. Denticle size and spacing was quantified by determining the number of denticles per 5 mm of carina length (see Farlow and Brinkman, 1987), counted as close to the base, mid point, and apex as possible (see Chandler, 1990), thus generating the basal (MB and DB), mid-crown (MC and DC), and apical densities (MA and DA). For very small teeth (e.g., Bambiraptor), counts were taken over 2 mm and then adjusted to 5 mm after Farlow et al. (1991). Five counts of each variable were taken, the means of which yielded average densities for the mesial and distal carinae (MAVG, DAVG, after Chandler, 1990). The ratio of MAVG to DAVG generated the denticle size density index (DSDI), devised and discussed by Rauhut and Werner (1995). Analyses Statistical analyses were run with SPSS, SigmaStat®, and Stat- View and were illustrated using SigmaPlot®. Factors that might have a significant effect on the variability within the data were examined using analysis of variance (ANOVA). As biogeography (concept after Carrasco, 2000a, b; Lieberman et al., 2002) showed no significant effect on the observed variation (Smith et al., 2005) and sexing and aging of tyrannosaurs is problematic (Larson, 1994; Chapman et al., 1997; Galton, 1999), tooth position is the main factor that might account for a significant proportion of the observed variation. ANOVA was employed to test this hypothesis using variability profiles sensu Yablokov (1974) but modified after Smith (2002) to show positional variation within tooth rows (sensu Williamson, 1996). As the variables were compared with tooth position rather than with coefficients of variation (see Yablokov, 1974), raw data were analyzed instead of following Sokal and Braumann (1980). The results were examined using post-hoc tests that compared the means of the dependent variables with respect to tooth position: Fisher’s PLSD (see Sokal and Rolf, 1995) and Tukey-Kramer (Kramer, 1956). Raw data were used for AL, CA, CBL, CBR, CBW, CH, CHR, MAVG, and DAVG. CA and DAVG were also compared after removing size as a confounding variable (see Marko and Jackson, 2001). Smith et al. (2005) log-transformed the data and ran a principal-components analysis. The data for DAVG were then regressed on PC1, which explained 84.4% of the observed variation; the residuals from these regressions constitute the sizecorrected variables CA2 and DAVG2. MORPHOLOGY AND POSITIONAL VARIATION Osborn’s (1912) study remains the best prior treatment of Tyrannosaurus dentition, although other works have briefly discussed the teeth (e.g., Brochu, 2002; Hurum and Sabath, 2003). Osborn (1912) focused on CM 9380, AMNH 5027, 5117, 5866, and CM 9379. He reported a dental formula of 4 premaxillary, 12 maxillary, and 13–14 dentary teeth, for 58–60 total positions. Molnar (1991) found 4/11–12/? from these specimens and LACM 23844 and SDSM 12047. Currie et al. (2003) reported 4/11–12/ 12–14, but from which specimens these data came is not clear. Table 2 provides an updated list of tooth counts. Theropods are not usually considered to be heterodont taxa, but this view is too simple (see Currie, 1987). Tyrannosaurus rex in particular exhibits as much heterodonty (see Stromer, 1934) as do taxa acknowledged as having distinctive dentitions, such as Eoraptor Sereno et al., 1993, Troodon, and Masiakasaurus. Preliminarily, T. rex appears to be more heterodont than Daspletosaurus or Albertosaurus; it possesses tooth morphologies that we can regard as classes at the element level (see Peyer, 1968; Zhao et al., 2000) and as sets sensu Hungerbühler (2000) at the intraelement level. These sets differ in concept from the ontogenetic tooth families of Edmund (1969) and Osborn (1973). Broadly, T. rex follows the ‘typical’ theropod pattern (see Holtz and Osmólska, 2004) of ‘recurved’ crowns possessing longer base lengths than widths (Molnar and Carpenter, 1989). However, T. rex teeth are less curved than those of many theropods (an obvious exception to this are the dentitions of certain spinosaurids; see Sereno et al., 1998; Sues et al., 2002). The Premaxillary Dentition There are four teeth in all known premaxillae of T. rex, a feature that is robust for the Tyrannosauridae (see Osborn, 1905, 1906, 1912; Carpenter, 1990, 1992; Molnar, 1991; Brochu, 2002; Currie, 2003a; Hurum and Sabath, 2003). Premaxillary tooth count is constant for most theropods (see Lamanna, 1998). Currie (pers. comm., 1998; see also Ji and Ji, 1997) noted a discrepancy in tooth count in a specimen of Sinosauropteryx Ji and Ji, 1996 (NIGP 127586), but the lack of variation in other theropods suggests that this might be a specimen-specific or taxon-specific anomaly (assuming the individuals all represent one species). Currie and Chen (2001) reported that preparation issues confound tooth counts in Sinosauropteryx. Baryonyx might possess six left and seven right premaxillary teeth in BMNH R9951 (see Charig and Milner, 1997). However, Rpm6 and 7 appear to be crowded into one alveolus, suggesting that BMNH R9951 could also be anomalous. Such anomalies are common in mammals (P. Dodson, pers. comm., 2004) and should not affect the stability of theropod counts. It is also possible that tooth-count variation increases with decreasing crown size (J. D. Harris, pers. comm., 2004). Crown Size —The premaxillary class is significantly smaller than the maxillary class in CBL, CBW, CH, and AL (Table 1, Fig. 2), as in other tyrannosaurids (Russell, 1970; Holtz, 2001; Currie, 2003a; Hurum and Sabath, 2003) and some ceratosaurs (Rauhut, 2004). There are trends of increasing size along the tooth row for CBL, CBW, CH, and AL (Fig. 2). The first tooth (mean pm1 CBL = 28.61 mm) is significantly smaller than that for pm3 (pm3 CBL = 33.96; ANOVA with pm1: F = 2.92, p.0346), but the remaining teeth are not significantly different from each other in size. The fourth tooth is slightly smaller than pm3 in all size variables, but the differences are not significant. A similar decrease in size toward the premaxillary-maxillary suture (premaxilla-maxilla joint sensu Molnar, 1991) occurs in Coelophysis Cope, 1889, Dilophosaurus, and Eoraptor (Welles, 1984; Colbert, 1989; Sereno and Novas, 1993). Crown Shape and Carina Morphologies —Theropod crown basal cross sections are often ovals that taper to points corresponding to the locations of the carinae (Fig. 1B). The basal long axis (generally oriented mesiodistally) can be twice the length of the short axis (usually oriented labiolingually). In the premaxilla of T. rex however, although the crown bases are narrow ovals, the long axes are oriented labiolingually (Fig. 1C). The labial faces are ovals that are convex toward the rostral end of the snout (Fig. 3A). The lingual faces form very weakly convex curves (e.g., AMNH 5027), which are more pronounced basally, creating a shallow wide ridge (contra Molnar and Carpenter, 1989); apically the faces are almost flat (Fig. 3B). The curves of the lingual faces flatten out proximal to the carinae, which are located at the mesiolingual and distolingual corners of the crowns. The flattening is more pronounced distal to pm1 where the very slightly convex mesial and distal faces curve into the lingual face. A number of theropods exhibit positional variation with respect to carina placement. Tyrannosaurus rex exhibits dramatic changes in this feature along the maxillary and dentary tooth rows, but virtually none within the premaxilla: the carinae are all located at the corners of the teeth, as in other tyrannosaurids (Russell, 1970; Currie, 2003a). In labial view, the denticles are often just visible along the mesial and distal edges of the labial faces. The carinae start from lingual points on the apices and extend basally along the mesial and distal faces to the bases, exhibiting slight labial convexity. They cross the apices with a surface expression of one carina wrapping over the tip rather than two distinct carinae, although each is discussed separately here. The mesial carinae are ̴2–5 mm shorter than the distal and often do not extend to the bases. Both carinae in pm3 of BHI 3033 terminate before reaching the base. Often premaxillary mesial carinae of T. rex are slightly shorter than the distal carinae, so it might be possible to discriminate left and right crowns by identifying the distal carina. Similar morphologies occur in the premaxillae of other tyrannosaurids (Russell, 1970; Carr, 1999; Currie, 2003a), leading to the premaxillary crowns being referred to as ‘incisiform,’ ‘U-shaped,’ or ‘D-shaped in cross section’ (see Russell, 1970; Currie et al., 1990; Carpenter, 1992; Holtz, 1994, 1998; Carr, 1999; Ford and Chure, 2001; Brochu, 2002; Currie, 2003a; Carr and Williamson, 2004). Given the convexity of the lingual face and the oval shapes of the crown bases, these descriptors are somewhat inaccurate, but do get the general meaning across (at least colloquially; better descriptions might be desired when using this feature as a systematic character). Although theropod teeth in general tend to be more circular mesially and more bladelike distally (see Smith et al., 2005), T. rex exhibits a reverse trend. The premaxillary teeth are significantly more blade-like than those in the maxilla or dentary (Table 1, Fig. 4A). This might be initially surprising, but T. rex premaxillary crown bases are very elongate. The first two teeth are significantly less circular than are than are pm3 and 4 (pm2 CBR = 0.57; pm3 = 0.64, F = 14.06, p.0128). The premaxillary teeth are more ‘squat’ than in the maxilla or dentary (Table 1, Fig. 4B). Along the tooth row, the crowns become increasingly, but not significantly, more elongated (CHR values of 1.52, 1.52, 1.64, and 1.74, from p1–pm4, respectively). The rostral end of the snout in T. rex is more squared off than in many theropods (Fig. 5A), so that the tooth row curves away from the midline. This, along with the derived crown shapes, produces the distinctive tyrannosaurid premaxillary dentition. Carina placement and lingual face morphology in T. rex are similar to those of pm1 of Allosaurus (e.g., YPM 1333 and 4933, MOR 693, and UMNH 1251) and pm1–3 in Majungatholus (e.g., FMNH PR2100). In all three taxa the carinae form the mesial and distal edges of the lingual faces in lingual view (Figs. 5 B–D). However, this resemblance of Allosaurus and Majungatholus premaxillary teeth with those of T. rex is only valid mesially. By the distal ends of the tooth rows in both of these taxa, the crowns are distinctly different from the premaxillary condition in T. rex. By pm4 in Allosaurus (YPM 1333), the mesial carina, in mesial view, begins at the apex and curves lingually such that it forms the lingual edge of the mesial face at about the mid-crown point. In pm5 the distal carina essentially defines the distal keel of the crown, showing only a very slight labial convexity along its length. Crown Curvature —That crown curvature is taxonomically variable in theropods has been suggested (Gilmore, 1920; Russell, 1970; Madsen, 1976). Within-taxon variation takes the form of increasingly curved crowns along the length of the tooth row (see Smith 2002), often accompanied by a decrease in size (Chandler, 1990). However, some taxa, such as Spinosaurus Stromer, 1915, and Irritator Martill et al., 1996, exhibit very little mesial curvature (see Sues et al., 2002). This curvature can be seen indirectly in the CA data generated by equation (1). In general, CA values closer to 90° indicate more centrally positioned apices, a condition that usually correlates with less curved mesial profiles. Lower CA values usually correlate with more strongly curved mesial profiles. In T. rex, curvature decreases from pm1–4 (Fig. 4C), but the differences are not significant (mean CA values of 84.7°, 85.1°, 85.9°, and 85.9° from pm1–4). The CA values in the premaxillary dentition versus those in the maxilla and dentary are also not significant (Table 1, Fig. 4C). Denticles —The mesial premaxillary denticles range in size from ̴9–̴11/5 mm (Table 1), with a weak trend of increasing size along the tooth row (Fig. 6A), although differences between adjacent teeth are mostly not significant. Premaxillary MAVG values are not significantly larger than those of the maxillary or dentary teeth (Table 1). In examining the components from which MAVG is generated, the MA data range from 8.4–10.9/5 mm, the MC data from 7–9.5/5 mm, and the MB data from 9–12.5/5 mm. There is no significant size trend along the tooth row in the apical denticles, but the mid-crown denticles show a very weak increasing trend. The MB data show no clear trends. The apical mesial denticles (10.3/5 mm) are significantly smaller than those of the maxilla (7.9/5 mm, p d12) are different from all other T. rex crowns; they are small, very narrow, and have strongly curved mesial curvature profiles and almost straight distal profiles that in some cases (Ld13 of BHI 3033) angle toward the apex. Even in the last few teeth, however, the mesial carinae exhibit a basal lingual twist and the distal carinae are labially placed (e.g., Rd13 of LACM 150167). As in the maxilla and in Acrocanthosaurus, but in contrast to the condition in Albertosaurus, Allosaurus, and Gorgosaurus, the mesial dentary teeth (e.g., MOR 008, AMNH 5027) are not set in the jaw with their long axes lined up mesiodistally. Rather, the crowns are oriented such that there is an angle of ̴30° between the long axes and the alveolar margin. Thus, the apices do not line up with each other along the tooth row and the surface area the teeth can contact during a bite is greater in T. rex than it is in other tyrannosaurids such as Gorgosaurus (see discussion). Crown Curvature —As in the maxilla, curvature increases across the dentary and position is a significant factor in explaining the observed variation (Fig. 4C). The CA for d1 (84.44°) is significantly different from those of the distal teeth (d11= 82.68°, p.0357). Whereas d1 might be successfully distinguished from pm1 using CBW, it is not distinct from pm1 in terms of apex displacement (pm1 = 84.72°, p.7154). However, pm1 has a more strongly curved mesial profile than does d1 (Fig. 15). As the crowns become more elongated in the mesial dentary, their apices become more centrally positioned and there is a weak trend of increasing CA in the set (Fig. 4C). However, in the distal set, CA values decrease steadily to the end of the row. The last dentary teeth have the most displaced apices in the dentition (d13 CA = 78.17°; d12 = 82.93°, p <.0001). Denticles —As in the rest of the mouth, denticle sizes in the dentary are reciprocals of crown size, and denticle density increases distally along the tooth row (Fig. 6). Mesial densities range from 7–14.4/5 mm and distal densities range from 7.5– 15.7/5 mm; d4 has the largest mesial denticles (7.8/5 mm) and d3 has the largest distal denticles (8.8/5 mm). The distal dentary teeth have the smallest denticles in the entire dentition (d13 DAVG: 14.9/5 mm; d12: 12/5 mm, p.0006). Contrary to Carr and Williamson (2004), there is no significant difference in denticle size between the mesial and distal carinae (MAVG = 9.6/5 mm; DAVG = 10/5 mm, p.5100). The DSDI data for the dentary (Fig. 6C) exhibit a similar trend to that observed in the maxilla; DSDI values do not increase or decrease substantially along the tooth row. NON-POSITIONAL VARIATION Apical Denticles —Carinae can be isolated or can cross the apex as one entity. For theropods with ‘single’ carinae, the denticles can terminate prior to or can cross the apex. This latter condition, apical denticulation, has been described in Acrocanthosaurus, Alectrosaurus Gilmore, 1933, Dryptosaurus Marsh 1877, Eotyrannus Hutt et al., 2001, Neovenator Hutt et al., 1996, and Ricardoestesia Currie et al., 1990, and is considered unusual (see Harris, 1998; Hutt et al., 2001). However, I have observed it in Majungatholus, Allosaurus, Carcharodontosaurus, tyrannosaurids, and dromaeosaurids, a distribution similar to that reported by Currie and Carpenter (2000). I agree with these authors that apical denticulation in and of itself is not a particularly useful character. However, recording whether or not the denticles cross the carinae (with possible taphonomic modification acknowledged) in theropod descriptions is still useful as I suspect that possession of apical denticulation is likely to be the plesiomorphic theropod state, while a lack of apical denticles is the derived condition. Crown Ornamentation —‘Enamel wrinkles’ are considered diagnostic for carcharodontosaurids (see Stromer, 1931; Sereno et al., 1996; Chure et al., 1999). Evaluating the distribution of the feature is beyond the scope of this paper, but observations made during this study indicate that the distribution of ‘enamel wrinkles’ extends beyond Carcharodontosaurus and Giganotosaurus Coria and Salgado, 1995, as similar structures occur in tyrannosaurids (including T. rex), allosaurids, Dromaeosaurus (AMNH 5356), and (very weakly) in Irritator (SMNS 58022). They are more strongly developed and exhibit slightly different morphologies in Carcharodontosaurus (SGM Din-1) as compared to other theropods, but ‘enamel wrinkles’ (broadly defined) are not restricted to this group, especially as there is no consensus as to what exactly constitutes an ‘enamel wrinkle.’ Split Carinae —This feature and its implications were discussed by Erickson (1995) and will not be restated here. Rather, as Erickson’s (1995) work focused on an assemblage of shed tyrannosaurid teeth, I will simply note the instances of in situ split carinae observed during this study. The first occurs in Rd2 of SDSM 12047, where an additional mesial carina occurs on the mesial face. It begins ̴12–13 mm from the apex, ̴1–2 mm labial of center, and extends down the labial quarter of the face. It is denticulate along its entire length; the denticles are smaller than those on the principal carina (̴10/5 mm apically; ̴9/5 mm basally). The second occurs in CM 1400, a partial left maxilla from the Lance Formation of Wyoming (see McIntosh, 1981). This bone possesses six complete alveoli and preserves four teeth. The third tooth back from the rostral margin (Lmx3?) has a secondary carina labial to the principal keel apically on the crown. These are the only instances of in situ split carinae in T. rex that I have observed. A third possible occurrence exists on the basal margin of the distal carina of Ld2 on BMNH R5863. However, there is a crack in the crown enamel between the two serrated ridges and it is possible that the crack runs through the distal carina, separating it and giving it the appearance of being split. Denticle Curves —Theropod denticle morphology has recently begun to be addressed (Currie et al., 1990; Slaughter et al., 1994; Baszio, 1997; Holtz, 1998; Holtz et al., 1998). Chisel-shaped denticles are said to be distinctive dromaeosaurid characters (Currie et al., 1990), pointed denticles that hook apically have been cited as characteristic for tyrannosaurids (Currie et al., 1990; Abler, 1992) and for troodontids (Currie, 1987). In reality, denticle morphology has only been examined at very basic levels (e.g., Currie et al., 1990) and variation in denticle shapes has not yet been examined. The true taxonomic and systematic utility of denticle shapes is currently unknown and applications of these shapes are premature. It might be profitable in the future to mathematically model the curved shapes of denticles as was discussed by Smith et al. (2005) for crown curvature profiles. DISCUSSION Teeth of T. rex possess several theropod plesiomorphies, including ‘sharp’ and ‘not closely packed’ crowns (Gauthier, 1986) with serrated carinae (Holtz, 1998) and significant curvature (Sereno et al., 1998). However, accounting for positional variation, T. rex displays features distinct from the teeth of other tyrannosaurids and non-tyrannosaurids that might hold promise for taxonomy and systematics. It is important to assess the distribution of these features within the Theropoda to gauge systematic potential (as is true for non-dental traits), and the strength of the analysis presented here is reduced without detailed comparisons of other dentitions. The discussion must begin somewhere, however, and the length of this article illustrates that the process of accounting for variation in theropod dentitions is involved enough to make the simultaneous study of nu- merous taxa impractical, especially those with heterodont dentitions. The discussion of putative characters given below is thus a very preliminary step that awaits detailed examinations of the dental arcades of other theropod species. Placing Teeth in Alveoli A stepwise discriminant analysis using squared Mahalanobis distances was run to study the prospect of correlating teeth with alveoli. The analysis was run in the same manner as those in Smith et al. (2005) and used AL, CA2, CBL, CBR, CH, CHR, CBW, and DAVG2 (see Smith et al., 2005, for data). The analysis succeeded in correctly classifying 41% of the teeth with the correct alveolus, not a good result. It is possible that sample size (116 valid teeth) is to blame. If so, the result is discouraging as this is a larger sample than is likely to be obtained for most theropods for the foreseeable future. However, results improve when tooth class, rather than position, is used as a factor (Table 3). In this analysis, 67% of the teeth were correlated with the correct bone. As it is difficult to discriminate T. rex maxillary and dentary crowns using visual inspection, these results offer some promise as to the future potential of being able to correlate teeth with alveoli, especially as more data are added to the standard. Taxonomy and Systematics Tooth Count — Lamanna (1998) found that premaxillary tooth count varies little within theropod species and is a robust character. The present work and all consulted sources agree that T. rex possesses the theropod plesiomorphy of four premaxillary teeth (Holtz, 1998). In general, maxillary and dentary counts are weak characters, except that high counts are synapomorphic for certain taxa, such as ornithomimids and some spinosaurids (but see Holtz, 2001; Currie, 2003a). Tyrannosaurids possess too much intraspecific variation in maxillary and dentary tooth counts for these features to be systematically useful (Lamanna, 1998). Tooth Emplacement —The en echelon emplacement of crowns that occurs in the lateral dentitions of some theropods (Fig. 11) is intriguing and it is tempting to interpret this feature in a systematic sense. A difference between the trend of the bone and the crowns’ basal long axes would result in the cutting of a wider swath during a bite than with teeth that line up along the trend of the bone, as occurs in some modern reptiles (see Auffenberg, 1981). This wider swath might serve to increase the efficiency with which a theropod could tear meat from a prey animal and as such there is likely to be a strong functional aspect to this feature. The en echelon emplacement of lateral teeth appears to be restricted to certain theropods and might represent a derived condition. The distribution of this feature within the Theropoda is curious, however (e.g., Acrocanthosaurus, Giganotosaurus, Majungatholus, T. rex), and warrants additional study. Tooth Size —Dental size features are not common in theropod phylogenetic analyses. Indeed, basal width is the only common theropod tooth size character (narrow crown bases are considered plesiomorphic for the Theropoda, Holtz, 2001). While size features must be handled carefully if they are to be examined in a systematic light, T. rex crowns are generally so much larger than those of other theropods in terms of CBL and CH that these features might be useful if compared with other metrics (e.g., limb lengths, Currie, 1998). Such appears to be the case for CBW. Although teeth of Carcharodontosaurus are similar in size to teeth of T. rex (Figs. 16A, C, D), T. rex crowns are substantially wider (Fig. 16B). In fact, T. rex might well have the widest teeth of any theropod, a feature affected less by position than some of the other variables (Fig. 2). Large basal width thus might ultimately prove to be a viable autapomorphy of Tyrannosaurus. Holtz (2001) erected a similar character (79: incrassate crowns: cross section greater than 60% wide labiolingually as long mesiodistally). In examining the data here, Holtz’s character (79) holds up for T. rex and Daspletosaurus, but is not robust for Gorgosaurus. Currie et al. (2003) discounted Holtz’s character 79 with the argument that base width is allometric and juvenile tyrannosaurids should possess ziphodont ‘lateral’ teeth. This is entirely possible, but Currie et al. (2003) did not support the hypothesis. There was no discussion of the data nor were juvenile crowns identified within the dataset. Although postulated juvenile crowns identified as cf. T. rex are known (pers. obs.; see also Currie et al., 1990), none of these has been conclusively shown to pertain to T. rex because there are currently no proven juvenile skulls of T. rex. For T. rex at least, the dentition of which differs from other tyrannosaurids in several ways, the lack of definitive juvenile data makes hypotheses regarding the ‘juvenile condition’ of the teeth of this animal speculative. The spread of the data reported by Currie et al. (2003:fig. 1) does not demonstrate that juvenile tyrannosaurids have ziphodont teeth, nor does it demonstrate that T. rex lacks a distinct CBW condition (various taxa possess aspects of their teeth that depart from clean linear relationships, see Farlow et al., 1991; Holtz et al., 1998). Carr and Williamson (2004), Carr et al. (2005), and Currie (2003a) reported that a ‘small’ mx1 is a tyrannosaurid character. Qualitatively, mx1 in Daspletosaurus and Gorgosaurus is smaller than the mesial maxillary dentition (see Carr, 1999; Currie, 2003a). In T. rex, however, although mx1 is the smallest crown in the mesial maxillary set, it is not ‘small’ with respect to the rest of the dentition. Indeed, mx1 is significantly larger than most of the maxillary teeth and almost the entire dentary class (Fig. 2), a feature recognized by Osborn (1912). A ‘small mx1’ is not a synapomorphy of the Tyrannosauridae, nor does it unite the Tyrannosaurinae sensu Currie (2003b) as the condition occurs in Daspletosaurus, but not in T. rex or in Tarbosaurus; this merits investigation given the relationships recovered by Holtz (2001) versus those obtained by Currie et al. (2003). Also intriguing is the fact that CMNH 7541 appears to possess a ‘small mx1.’ If additional material of ‘ Nanotyrannus ’ continues to support the possession of this feature, it might have implications for the taxonomic validity of this taxon. LACM 12471 also possesses a ‘small mx1,’ which might have implications for the hypothesis of Carr and Williamson (2004) that this specimen represents a juvenile T. rex. Even if tooth size in the maxilla of T. rex increases with positive allometry (Currie, 2003b), there is little reason to believe that mx1 would respond differently during growth than the rest of the maxillary class, especially as it is possible that theropod teeth exhibit small amounts of ontogenetic change (see Currie et al., 1990). As such, given the normal caveats related to sample size, the disproportionately small sizes of mx1 in some tyrannosaurids are probably real features; while not a tyrannosaurid synapomorphy, it might still be useful in helping to recover relationships within the clade. Carr and Williamson (2004) reported that d1 in tyrannosaurids is ‘subconical’ in shape and is the smallest tooth in the dentary. Their precise definition of ‘subconical’ is difficult to ascertain, but as discussed above, the morphology of d1 in T. rex is certainly different from the rest of the dentary teeth and indeed is similar to the premaxillary dentition. However, although d1 is similar in size to d11 (Fig. 2), it is significantly larger than the distal teeth of both the dentary and the maxilla (d12–d14; mx12). For at least T. rex, then, a ‘small’ d1 is not a viable character. Tooth Shape —‘Incisiform’ premaxillary crowns with linguomesially and linguodistally placed carinae are generally considered to be derived for tyrannosaurids (see Carpenter, 1982; Bakker et al., 1988; Molnar and Carpenter, 1989; Chandler, 1990; Currie et al., 1990; Abler, 1992; Farlow and Brinkman, 1994; Carr, 1999; Carr and Williamson, 2000; Sankey, 2001; Carr and Williamson, 2004). The condition is real, although some other theropods (e.g., dromaeosaurids) also possess premaxillary features that are similar to what occurs in tyrannosaurids (pers. obs.; see also Molnar, 1978; Molnar and Carpenter, 1989; Holtz, 1998). There might be an alternative to coding premaxillary crowns as simply “D-shaped in cross section,” to better distinguish the tyrannosaurid condition. In contrast to other theropods, tyranno- saurid premaxillary basal long axes are labiolingually oriented, with carinae that are positioned at the linguomesial and linguodistal corners of the crown. In the premaxillary dentitions of Allosaurus and some other taxa, the long axes are not strictly labiolingual and the placement of the carinae is more complicated. As such, tyrannosaurid premaxillary mesiodistal axes are distinctly shorter than those of other theropods; this is a derived condition of the clade (see Fig. 5). As with crown size, an ‘incisiform’ mx1 has been described as a tyrannosaurid synapomorphy (Carr, 1999; Currie, 2003a; Carr and Williamson, 2004), occurring in Gorgosaurus and Daspletosaurus as well as in problematic specimens such as LACM 28471 and FMNH PR2211 (e.g., Molnar, 1978; Carr and Williamson, 2004). However, the premaxillary teeth of T. rex and other tyrannosaurids have also been referred to as ‘incisiform’ in shape (e.g., Russell, 1970; Holtz, 1998; Brochu, 2002). In T. rex, mx1 is morphologically distinct from the more distal teeth and from the premaxillary class. The difference in shape is more dramatic between mx1 and the premaxillary class than it is between mx1 and the rest of the maxillary series; mx1 does not possess the same morphology as the premaxillary crowns and the term incisiform cannot be used in the same sense for both pm4 and mx1 in T. rex. The long axis of mx1 is mesiodistally oriented and is longer than the labiolingual axis. The mesial carina extends down the lingual side of the mesial face, which is the homologue to the labial face in the premaxilla. This is not the morphology of mx1 in Daspletosaurus or in Gorgosaurus and it is distinctly different from the premaxillary condition in T. rex. I would argue that there is no single good term for the morphology of mx1 in T. rex and that this crown is best described as a transitional form between the premaxillary and the maxillary classes, but one that is decidedly more similar to the maxillary condition. Using ‘incisiform’ in the same sense for both morphologies is not appropriate; if an ‘incisiform’ mx1 occurs in Gorgosaurus and Daspletosaurus, then the condition that actually occurs in T. rex is a lack of an incisiform mx1. CMNH 7541, the holotype of Nanotyrannus, and LACM 28471 both possess an ‘incisiform’ mx1 sensu Carr (1999). If ‘incisiform’ is describing the same morphologies in these non- T. rex taxa, then the presence of an ‘incisiform’ mx1 in CMNH 7541 and LACM 28471 might argue for the distinction of these specimens from T. rex and against the juvenile T. rex hypotheses advocated by Carr (1999) and Carr and Williamson (2004); this would be curious because the evidence offered by these authors is very compelling. Carina Lengths —In T. rex, the mesial carinae terminate above the crown bases and the distal carinae extend to the bases. Currie and Dong (2001) reported that the mesial carinae of maxillary teeth extend to the crown bases, but the results presented here refute this hypothesis. Shortened maxillary carinae occur in several theropods (e.g., Gorgosaurus, Acrocanthosaurus). Carr and Williamson (2004) reported that this condition is typical of tyrannosaurids except Daspletosaurus. In T. rex it is possible to interpret this feature as a trend of decreasing mesial carina length along the maxillary tooth row; in the distal crowns the mesial carinae terminate ̴25 mm above the bases of the enamel. Carina length cannot be used to discriminate a maxillary crown versus one from the dentary, but the fact that some theropods do not appear to possess shortened maxillary carinae (e.g., Velociraptor) suggests that this feature holds some taxonomic utility. Denticles —Since denticle and tooth sizes generally scale together (Chandler, 1990; Farlow et al., 1991; Baszio, 1997), denticle size would seem unlikely to be useful in taxonomy or systematics. However, few tests have been done and denticle sizes are occasionally considered to be taxonomically diagnostic (Molnar and Carpenter, 1989). Chandler (1990) reported that the taxonomic value of serration densities had not been assessed prior to her work because of a lack of diagnostic specimens (Carr, 1999, and I have had similar problems). Assessing the utility of denticles has been further hampered because existing published data are often calculated from a single tooth or as an average of several crowns (e.g., Barsbold, 1983; Currie, 1995; Azuma and Currie, 2000; Currie and Carpenter, 2000; Hutt et al., 2001). There is sometimes no distinction made concerning from which carina or even which tooth measurements come (e.g., Barsbold, 1983; Hurum and Sabath, 2003). Chandler’s (1990) lamentation over a lack of data reflected the additional problem that denticle densities have virtually never been reported in such a way as to facilitate a detailed examination of their variability. Denticle sizes alone do a poor job of discriminating most theropods (Figs. 17A, B). Rauhut and Werner (1995) devised DSDI to improve this situation, but the results are mixed (Fig. 17C). It discriminates poorly overall, but Deinonychus is significantly different from Dromaeosaurus, two taxa with otherwise very similar teeth (see Currie et al., 1990). This is mostly due to differences in MAVG between these taxa. There are other potentially useful tooth and denticle relationships as well. Troodontids and dromaeosaurids have derived teeth (Holtz, 1998, 2001) with significantly larger distal than mesial denticles (contrary to Carr and Williamson [2000; 2004], this is not typical of tyrannosaurids, at least where T. rex is concerned) and some spinosaurids have unusually small denticles for the size of the teeth. DAVG and DSDI values do not illustrate these characteristics well, but some resolution comes from using the size corrected DAVG2 (Fig. 17D). Mean DAVG2 values below ̴−.5 might be significant for troodontids and above.5 might be a synapomorphy for baryonychines. A tooth/denticle size index might thus generate a useful theropod character. Although devising this is beyond the scope of this paper, the concept has been qualitatively discussed previously by Sereno et al. (1998) and was explored by Farlow et al. (1991). Denticle size should be explored further. Summary Theropod teeth are simple structures, but this work has shown that there can be useful information contained within theropod dentitions if they are studied in detail, by combining qualitative descriptions with quantitative methods. With the dental anatomy and variation of a common theropod documented, we might ultimately expect additional systematic information to come from teeth. Also, a standard now exists against which to compare putative teeth of T. rex. It should now be possible to examine assemblages of isolated crowns from Upper Cretaceous rocks in western North America and identify teeth of T. rex within these assemblages. Descriptions of other dentitions should facilitate the inclusion in theropod phylogenetic analyses of additional dental information, facilitate the taxonomic identification of isolated teeth, and aid in assessing the validity of ‘tooth taxa,’ benefiting all biogeographical and paleoecological research conducted in terrestrial Mesozoic rocks.

Published as part of Joshua B. Smith, 2005, Heterodonty in Tyrannosaurus rex: Implications for the taxonomic and systematic utility of theropod dentitions, pp. 865-887 in Journal of Vertebrate Paleontology 25 (4) on pages 866-884, DOI: 10.1671/0272-4634(2005)025[0865:HITRIF]2.0.CO;2, http://zenodo.org/record/3942994