## 2 Results
### Growth Associations between Root Types Vary under Each Treatment  
The relationship between the number of 1 st-order lateral roots and the length of the primary roots where they branched out from showed different correlations (Figure 5a). When we analyzed the fitted line per treatment, the association that followed was an exponential relationship between the two given variables. In general, there was an exponential increase in the number of emerging 1 st-order laterals with the growth of the primary roots. The strongest association was found in the control plants at R\({}^{2}\) = 0.9165, which was followed by the inoculated plants at R\({}^{2}\) = 0.8688, which were both under ambient conditions. The plants that were subjected to the high temperature showed lower correlations, with that of the control plants being R\({}^{2}\) = 0.814 and the weakest one from the inoculated plants being R\({}^{2}\) = 0.3516 (Figure 5a). Figure 4: **Root length distribution across different depths through time. ** Length distribution of plant roots from all of the treatments across different depths or layers (20 horizontal sections) at  
several timepoints (9, 12, 14, 16, 19, and 21 DAI) for total root length (**a**); First order lateral roots (**b**) Second order lateral roots (**c**). Treatments: (black points and open symbols)--Amb-Ctl (control plants under ambient condition), (black points, closed symbols)--Amb-PsJN (PsJN-inoculated plants under ambient condition), (red points, open symbols)--HT-Ctl (control plants under high temperature), and (red points, closed symbols)--HT-PsJN (PsJN-inoculated plants under high temperature). All of the points are the mean \(\pm\) standard error of \(n\)= 12 (Amb-Ctl), 15 (Amb-PsJN), 12 (HT-Ctl), 8 (HT-PsJN) samples within each treatment. Figure 5: **Correlation analysis for lateral roots**. (**a**) Correlation between the length of the primary roots (Figure 2b) and the number of 1 st-order lateral roots (Figure 2e). (**b**) Correlation between the number of 1 st- and 2 nd-order lateral roots of plants from all of the treatments. Treatments: Amb-Ctl (control plants under ambient condition)—grey, empty circle; Amb-PsJN (PsJN-inoculated plants under ambient condition)—black, filled circle; HT-Ctl (control plants under high temperature)—pink, empty triangle; and HT-PsJN (PsJN-inoculated plants under high temperature)—red, filled triangles. Trendlines follow similar colors as their corresponding treatments. Points were taken at different time intervals (5, 7, 9, 12, 14, 16, 19, and 21). A polynomial trendline was the best fit for the relationship between the number of 2 nd-order laterals and the corresponding number of 1 st-order laterals (Figure 5b). Although they are showing an increase through time, the number of both of the lateral roots did not increase at a constant rate. As with the relationship of the variables in Figure 5a, the highest association was also found in the ambient control plants (R\({}^{2}\) = 0.8308), while the lowest one was found for the heat-stressed inoculated plants (R\({}^{2}\) = 0.4925) (Figure 5b). The strong correlation for the different roots of the inoculated plants under the ambient condition (Amb-PsJN) indicates the already established beneficial effects of the bacteria PsJN for the unstressed plants, however, the low correlation that was found in the heat-stressed inoculated plants suggests that there were changes in the dynamics of the growth promotion. Bacterial Inoculation under Different Temperatures Influences the Leaf Area and Dry Weights with Different Magnitudes  
The high temperature caused a decrease in the projected leaf area commencing at 14 DAI (Figure 6a). Overall, the high temperature caused a reduction of 51% in the shoot dry weights of the control plants and a 47% reduction in this in the inoculated plants (compared to the ambient control plants) (Figure 6b). Figure 6: **Projected leaf area and dry weight.