## 5 Conclusions  
A 3D living stump slope model was established by using finite element software. The variable cross-section embedded beam elements were used to simulate the root structure of living stumps. The mechanical characteristics of living stump slopes with different taproot lengths were studied, suggesting that root systems in different growth periods had different reinforcement effects on slopes. The more developed the root system, the better the reinforcement effect on the slope. The tap roots in the middle and lower part of the slope mainly exert their tensile strength to achieve the effect of anchoring the soil.  
The maximum shear stress region of soil moved towards a deeper soil layer because of the existence of roots. The longer the taproot, the better the anchorage. At the same time, the existence of the root system prevented the shear stress transfer along the slope and improved the problem of soil shear stress concentration at the slope foot. The model only studied the simplified root structure. Thus, in the future, more complex root structures should be analyzed in the model and combined with more realistic soil distribution. Nevertheless, our model is very useful for studying the mechanical characteristics of living stump slopes, which can help us to understand how to optimize the type of root structure in ecological slope protection engineering and select good tree species to protect slopes more effectively. For example, planting more live stumps with developed taproots and lateral roots growing towards the slope in the middle and lower part of the slope will improve slope stability. Different planting schemes can be added to study how to arrange live tree stumps to make the slope most stable.  
In particular, under the influence of different factors such as rainfall, planting, and traffic load, the study of the mechanical characteristics of living stump slopes will help us to better apply the living tree stump slope protection technology.  
Conceptualization, X.J. and H.Y.; methodology, W.L. and X.J.; software, W.L. and H.Y.; formal analysis, X.J., Z.L. and W.F.; data curation, F.W.; writing--original draft preparation, W.L.; writing--review and editing, Z.L. and W.F. All authors have read and agreed to the published version of the manuscript.  
The key construction discipline research ability enhancement project of Guangdong Province(Nos.2022 JJ31005); The National Natural Science Foundation of China (Nos.31971727); The science and technology innovation Program of Hunan Province (Nos.2022 NK2056); The Forest Science and Technology Innovation Program of Hunan Province (Nos.XLK202105-3); The Special Fund of Safety Production Prevention and Emergency Response of Hunan Province in 2021 (Nos.2021-QYC-10008-24956); Hunan Provincial Natural Science Foundation of China (Nos.2022 JJ31005); The Key Project of Chongqing Natural Science Foundation(CSTC20)CYJ-ZDXMX0012); Open Fund of Engineering Research Center of Catastrophic Prophylaxis and Treatment of Road & Traffic Safety of Ministry of Education (Changsha University of Science & Technology: KFJ190401).  
Not applicable.  
The data presented in this study are available on request from the corresponding author.  
The authors declare no conflict of interest.