### Study area and data collection

The Jade Dragon Snow Mountain and its adjacent region (26°35′N–27°45′N, 99°22′E–100°32′E) are located in the northwest of Yunnan Province, China, bordering the Tibetan Plateau (Fig. 1). This region covers a total area of 6127 km^{2} and has a large altitudinal gradient from 1350 to 5050 m. The climate in this region is under the control of the southwest monsoon from the Indian Ocean, with richer precipitation in the eastern part than in the western part (Wang et al. 2007a, b). Seventy percent of the MAP occurs between May and September, with winter precipitation only contributing 30 % of the annual total. The Jade Dragon Snow Mountain, with its adjacent region, is a global biodiversity hotspot (Myers et al. 2000), where the flora is rich, including a total of 2028 native seed plant species from 625 genera and 136 families. More than half of the species are herbaceous (66.4 %; 82 families; 426 genera; and 1346 species), and 33.6 % are woody species (290 tree species from 120 genera and 56 families, and 392 shrub species from 133 genera and 62 families; Wang et al. 2007a, b). The vegetation types on the Jade Dragon Snow Mountain from low to high elevations are tropical forest, subtropical forest, alpine meadow and alpine tundra, which corresponds to the general altitudinal pattern of plant diversity in northwestern Yunnan Province.

We generated a species database from the published book *Checklist of Seed Plants of Lijiang Alpine Botanic Garden* (Wang et al. 2007a, b), which is based on substantial surveys on the Jade Dragon Snow Mountain and in its adjacent region. This species database includes minimum and maximum elevations of occurrence for each species, and life-form (tree, shrub, and herb) information on each species. We extracted topographical data from ASTER GDEM V2 (Global Digital Elevation Model Version 2, DOI:10.5067/ASTER/ASTGTM.002) with 30 m × 30 m resolution. We obtained MAT and MAP data from the WorldClim database (Hijmans et al. 2005), which is frequently used in ecological studies (e.g., Sommer et al. 2010).

### Equal-elevation altitudinal gradient

We divided the Jade Dragon Snow Mountain into 37 100-m altitudinal bands from 1350 to 5050 m. Along this equal-elevation gradient, we calculated the area of each altitudinal band by multiplying 900 m^{2} by the number of digital elevation model (DEM) grids in each band. The area increases steeply with increasing elevation, and then decreases above the 2650–2750 m altitudinal band, showing a hump-shaped pattern (Fig. 2a). MAT declined monotonically with elevation (Fig. 2b). MAP illustrated a much more complex pattern with elevation, increasing below 1900 m, then declining steeply to 912.0 mm at 4200 m and afterward increasing to the top of the Jade Dragon Snow Mountain (Fig. 2c).

We assumed that each species had a continuous distribution range between its recorded minimum and maximum elevations, as widely used in previous studies (e.g., Rahbek 1997; Vetaas and Grytnes 2002; Sanders 2002). However, among the 2028 seed plant species, there are 717 species that have been recorded only in a single elevational band. We counted the number of seed plant species, genera, and families present at each altitudinal band as observed species, genus and family richness (Sobs, Gobs, and Fobs). To explore altitudinal patterns of plant diversity for different life-forms, we counted the number of tree, shrub, and herb species occurring at each altitudinal band as the species richness for trees, shrubs, and herbs (TSobs, SSobs, and HSobs). Additionally, in order to evaluate whether the altitudinal biodiversity patterns and their determinant predictors were dependent on species’ range size, we divided all species into three groups according to their altitudinal range sizes: <150 m (Group I, 830 species), between 150 and 500 m (Group II, 554 species), and >500 m (Group III, 644 species). These range size limits were chosen to make the number of species in each group comparable. In the same way, we obtained the species richness for Groups I–III (ISobs, IISobs, IIISobs) by counting the number of species in Groups I, II, and III present in each altitudinal band. We thus generated nine observed taxon richness variables (hereafter TRobs): Sobs, Gobs, Fobs, TSobs, SSobs, HSobs, ISobs, IISobs, and IIISobs.

### Effect of interpolation

Like many previous studies, we considered a species to be present at all elevations within its recorded elevation limits (Kluge et al. 2006). However, this approach may produce artificially elevated species richness at mid-elevations, since such interpolated data are disproportionately added to mid-elevations as opposed to edges of the gradient (Karger et al. 2011). In order to find out whether using interpolated species richness masks its real pattern along the altitudinal gradient, we evaluated the relationship between elevation and species richness of a particular plant species set (717 species that have been recorded only in a single elevational band) without interpolation (Vetaas and Grytnes 2002). We checked whether this species richness pattern generated without interpolation manifested a hump-shaped curve. If it indeed shows a hump-shaped pattern, we can conclude that the effect of interpolation may not be essential for the hump-shaped pattern of plant species on the Jade Dragon Snow Mountain.

### Species–area relationship

The species–area relationship has been universally acknowledged, but the exact structure of the relationship is still under discussion (Connor and McCoy 1979; Crawley and Harral 2001). Three common versions of the species–area relationships are: untransformed (species richness versus area), semi-log (species richness versus log area), and log–log (log species richness versus log area) transformed (Matthews et al. 2014). We conducted linear fittings to all three versions using ordinary least square (OLS) regression models, which was commonly used for studies on the species-area relationships (Matthews et al. 2014). To select the version (i.e., untransformed, semi-log or log–log) with the best performance, we calculated the modified Akaike information criterion (AIC_{C}) corrected for small samples as follows:

$$AIC_{C} = - 2 \times \log {\text{Lik}}\left( {\text{model} } \right) + 2K\frac{n}{n - K - 1},$$

(1)

where *n* is the number of samples and *K* the number of parameters in the model. We considered the model with the lowest AIC_{C} score to be the best model, which is consistent with previous studies (Matthews et al. 2014). Then we compared the difference between the AIC_{C} of each model and the minimum AIC_{C} found, and we refer to this difference as ∆(AIC_{C}). Any model with ∆(AIC_{C}) < 2 is reported to be as good as the best model (Burnham and Anderson 2002). This analysis was performed using the function ‘AICc’ within the ‘AICcmodavg’ package (Mazerolle 2011) in R 2.14.2 (R Core Team 2014). To further assess the performances of linear fits, we also measured the adjusted coefficients of determination and conducted the significant F-test (Crawley 2002).

The log–log transformed species–area relationship was found to be the best-fit model (Δ(AIC_{C}) > 2) for all plant groups (Additional file 1: Table S1; Figures S1–S3). We thus chose the log–log transformed version (i.e., power-law) to correct TRobs. This power-law model, S = *c*A^{z}, where S is TRobs, *z* is a constant describing the slope of the species–area relationship in the log–log transformed space (Additional file 1: Fig. S3), A is the area of elevational bands along the equal-elevation gradient and *c* is the area-corrected taxon richness (hereafter TRcor_{1}). In order to rescale TRcor_{1} to similar values as TRobs, TRcor_{1} were calculated as 100(TRobs/A^{z}).

### Equal-area altitudinal gradient

Another approach to account for area effect was introduced by Bachman et al. (2004), and we refer to this as method 2 in our analysis. The original DEM cell values are integers and were added a random number between −0.5 and +0.5, as done in previous studies (Bachman et al. 2004). This produced DEM cells are easily classified into 37 equal-area altitudinal bands from 1350 to 5050 m using software ArcGIS version 9 (Environmental Systems Research Institute, Redlands, CA, USA). The middle elevations of the equal-area bands were not uniformly distributed along the altitudinal gradient (Fig. 3). The altitudinal band width first decreased then increased as elevation rose (Fig. 4a). The band with the smallest width (35.4 m) was the 15th, while the largest width was 1014.1 m for the 37th band. MAT still declined monotonically, and MAP showed a similar but simpler pattern along the equal-area gradient compared with the equal-elevation gradient (Figs. 2b, c, 4b, c). Along the equal-area altitudinal gradient, we counted the number of seed plant species at each band as Scor_{2}. In the same way, we got Gcor_{2}, Fcor_{2}, TScor_{2}, SScor_{2}, HScor_{2}, IScor_{2}, IIScor_{2}, and IIIScor_{2} for other plant groups. All these nine area-corrected taxon richness variables were collectively called TRcor_{2}.

### Mid-domain effect

We used RangeModel (Colwell 2008) to generate the simulated taxon richness, which are the mid-domain null model predictions. This software placed empirical species ranges within the domain (i.e. the mountain range from the lowest elevation to the peak) randomly and under the constraint that no species extended beyond domain boundaries (Colwell and Lees 2000; Colwell et al. 2004). Then the number of species was counted within each elevational band. We conducted 1000 simulations, and the mean of these simulations was called a prediction from MDE. This procedure was carried out along the equal-elevation and equal-area gradient.

### Statistical analysis

We first conducted OLS regressions between taxon richness (TRobs, TRcor_{1}, and TRcor_{2}) and each explanatory variable (elevation, MAT, MAP, and prediction from MDE), respectively (Additional file 1: Figures S4–S6). In order to compare the performances of first- and second-order polynomial regressions, we calculated ∆(AIC_{C}) according to Eq. (1). Model with the minimum AIC_{C} and ∆(AIC_{C}) > 2, was selected as the best one. Two models with ∆(AIC_{C}) < 2 were considered to have the same good performance, and in this case, the first-order polynomial regression was selected as the best model for simplicity (Additional file 1: Tables S2–S4).

Secondly, we examined the spatial autocorrelation in the residuals of the best OLS model using Moran’s *I* coefficient. We regard OLS model residuals in the 37 sequential elevational bands (from 1350 to 5050 m) as 37 observations along a single geographic axis. Moran’s *I* coefficients are computed from pairs of observations found at preselected distances: distance = 100 m, 200 m, 300 m, etc. (Legendre and Legendre 1998). Moran’s *I* coefficient is defined as follows:

$$I = \frac{n}{S}\frac{{\sum\nolimits_{i = 1}^{n} {\sum\nolimits_{j = 1}^{n} {w_{ij} (x_{i} - \bar{x})(x_{j} - \bar{x})} } }}{{\sum\nolimits_{i = 1}^{n} {(x_{i} - \bar{x})^{2} } }},$$

(2)

where *n* is the number of elevational bands, *x*
_{
i
} and *x*
_{
j
} represent observations in elevational bands *i* and *j*, \(\bar{x}\) is the mean of all *x*, and *w*
_{
ij
} is an element in the (*n* × *n*) weighting matrix *W*. It can be given as follows:

$$w_{ij} = \left\{ {\begin{array}{*{20}l} 1 \hfill &\quad {i,j\,{\text{is found at a given distance}}} \hfill \\ 0 \hfill &\quad {\text{otherwise}} \hfill \\ \end{array} } \right.,$$

(3)

*S* represents the sum of the weights *w*
_{
ij
} (i.e., the number of connections in the matrix *W*) as follows:

$$S = \sum\limits_{i = 1}^{n} {\sum\limits_{j = 1}^{n} {w_{ij} } } ,$$

(4)

Moran’s *I* coefficient varies between −1.0 and 1.0 for maximum negative and positive spatial autocorrelation, respectively. Non-zero values of Moran’s *I* coefficient indicate that observations in elevational bands connected at a given distance are more similar (positive autocorrelation) or less similar (negative autocorrelation) than expected by chance. Through plotting Moran’s *I* coefficients against the preselected distances, we constructed spatial correlograms and detected considerable spatial autocorrelation in OLS model residuals (Additional file 1: Figures S7–S18). This analysis was performed using function ‘correlog’ within the package ‘ncf’ (Bjørnstad 2006).

However, spatial autocorrelation has been reported to inflate Type I errors and thus lead to the biased model comparison and poor parameter estimates, through violating assumptions of independence and identical distribution of model residuals (Dormann et al. 2007). Therefore, thirdly we ran simultaneous autoregressive (SAR) models of the error type (Kissling and Carl 2007) to correlate taxon richness (TRobs, TRcor_{1}, and TRcor_{2}) and each explanatory variable (elevation, MAT, MAP, and prediction from MDE), respectively. Spatial weights matrices in SAR were based on row standardization and neighborhood distance of 100 m. Pseudo-*R*
^{2} values of SAR were calculated as the squared Pearson product-moment correlation coefficient between observed and predicted values (Kissling and Carl 2007). We implemented SAR models with package ‘spdep’ (Bivand 2014). We also performed first- and second-order polynomial SAR models and compared their performances using ∆(AIC_{C}). The selection criterion of the best model was exactly the same as OLS analysis. The best SAR models reduced spatial autocorrelation in the model residuals to a lower level, especially at the distance of 100 m (Additional file 1: Figures S7–S18).

There is no consensus on which area-correction method is better. Each method has its advantages and drawbacks. In this study, we try to find which area-correction method preserves the influences of other factors to a larger degree after eliminating the area effect. Therefore, fourthly we compared SAR pseudo-*R*
^{2} values of method 1 and 2, and consider the method with the larger pseudo-*R*
^{2} values to have the better performance.