## 3 Results
### QTL Mapping of the Root System  
According to EDIE, the root system's growth can be considered a collectively controlled system using parameters \((K_{1},r_{1},\alpha,c_{1},d_{1},K_{2},r_{2},\beta,c_{2},d_{2})\). Based on Equation (5), we identified 93 significant loci under control group conditions (Figure 3a) and 44 significant loci under salt-stress conditions (Figure 3b) using a threshold of 0.5 \(\times\) 10\({}^{-3}\) after applying the BH method for FDR correction. These loci regulate the growth of both the taproot and lateral roots, as well as the influences of the environment and their interactive relationships. Tables S2 and S3 present the fundamental details of these QTLs, along with annotations on gene functions pertinent to _Populus euphratica_'s development or homologous genes critical to the development of other tree species. Taking Q2762 (nn_np_3833, Linkage Group 4) as an example, it is linked to a gene encoding a putative F-box/kelch repeat protein. The F-box proteins, constituting a diverse family, are linked to various biological functions in plants [48]. Within this diverse family of proteins, the kelch repeat F-box (KFB) protein subpopulation is an important class. KFB proteins in plants play a role in regulating circadian rhythms, promoting plant development, and regulating secondary metabolic processes [49]. Q1765 (hk_hk_2098, Linkage Group 3) is associated with a gene responsible for encoding E3 ubiquitin-protein ligase HERC2, a key regulator in the cellular DNA-damage-repair process [50].  
In the comparison across different environments, the differences in significant loci are evident in their positions within the linkage groups. Under control conditions, significant QTLs predominantly occur within Linkage Groups 1 (28%), 10 (13%), and 14 (17%). In saline-stress conditions, significant QTLs predominantly occur in Linkage Groups 4 (23%), 10 (11%), 12 (16%), and 13 (14%). The differences in significant loci indicate that under different environmental conditions, genes regulating the development strategies of the taproot and lateral roots vary, which also highlights the adaptive strategies of root system growth and development in response to environmental stress. Additionally, two QTLs, Q2434 and Q8011, are found to significantly influence root development across both control scenarios and saline environments (Figure 3c,d). Q2434 (hk_hk_399, Linkage Group 4) encodes uncharacterized LOC110037592. Q8011 (nn_np_6308, Linkage Group 18) encodes LOC105128035, corresponding to a pentatricopeptide repeat-containing (PPR) protein similar to At2g33760. PPR proteins serve numerous essential functions across the life cycle of a plant. They are particularly vital in the processing of RNA within the mitochondria and chloroplasts of plant cells [51].  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & \multicolumn{3}{c}{**Taproot–Lateral-Root**} \\ \hline \multicolumn{2}{c}{**Control**} & \multicolumn{3}{c}{**Saline Stress**} \\ \hline
**Lateral Root** & **Taproot** & **Lateral Root** & **Taproot** \\ \hline
31.3696 & 19.0193 & 33.4926 & 22.4966 \\ (\(K_{1}\)) & (\(K_{2}\)) & (\(K_{1}\)) & (\(K_{2}\)) \\ \hline
0.0815 & 0.0743 & 0.0923 & 0.0368 \\ (\(r_{1}\)) & (\(r_{2}\)) & (\(r_{1}\)) & (\(r_{2}\)) \\ \hline \(-\)0.0007 & 0.0004 & \(-\)0.0028 & \(-\)0.0007 \\ (\(\alpha\)) & (\(\beta\)) & (\(\alpha\)) & (\(\beta\)) \\ \hline
0.0215 & 2.3801 & 0.0221 & 0.8944 \\ (\(c_{1}\)) & (\(c_{2}\)) & (\(c_{1}\)) & (\(c_{2}\)) \\ \hline
0.0569 & \(-\)0.1686 & 0.0333 & \(-\)0.0742 \\ (\(d_{1}\)) & (\(d_{2}\)) & (\(d_{1}\)) & (\(d_{2}\)) \\ \hline \multicolumn{2}{c}{RSS =0.8597} & \multicolumn{3}{c}{RSS =0.5530} \\ \hline \multicolumn{2}{c}{R\({}^{2}\) =0.9997} & \multicolumn{3}{c}{R\({}^{2}\) =0.9994} \\ \hline \hline \end{tabular}
\end{table}
Table 1: The estimated parameters of the taproot and lateral roots fitted based on EDIE and goodness of fit (R\({}^{2}\)) and residual sum of squares (RSS).