## 4 Conclusions  
This research presents a comprehensive analysis of utilizing forest trees as natural mechanical metamaterials for reducing railway-induced ground vibrations. A three-dimensional semi-analytical model was developed to compute the dynamic responses of the coupled track-ground-tree system. It should be emphasized that the tree trunk model is simplified and does not account for the mechanical contribution of roots and branches, which can significantly influence the vibration behavior of trees (future work should progressively incorporate these complexities).  
A systematic evaluation was conducted on the performance of tree grids in mitigating railway-induced ground vibrations. In addition, the performance uncertainty resulting from the random variations in tree properties was assessed using Monte Carlo simulations. The conclusions are as follows:1. Tree grids produce considerable attenuation performance for train loads. Due to the interaction of wave fields excited by multiple harmonic point loads at different locations, the attenuation efficiency varies significantly within 5-50 Hz across surface positions, with observation points farther from the tree grid exhibiting lower attenuation efficiency. Moreover, the attenuation performance tends to weaken by 1-3 dB with an increasing number of carriages.
2. A wider tree distribution area, especially along the direction of the track, leads to more effective vibration mitigation, and the differences in ARS can reach 12 dB at farther points. Compared with the ordinary tree grid, trees arranged with a height gradient in the direction perpendicular to the track can capture Rayleigh waves induced by train loads and convert them into bulk shear waves, thereby achieving more significant attenuation performance within 10-63 Hz.
3. Under random variations in tree parameters (height, radius, and density) with a \(\pm\)40% fluctuation range, height induces the largest uncertainty, with ARS fluctuations up to 1.7 dB at close points in the 8-16 Hz range; radius has minimal effect, with variations under 1.2 dB at most; when all parameters vary simultaneously, the uncertainty can reach 2 dB (low-frequency) and 1 dB (high-frequency), but the grid still maintains attenuation performance within 8-32 Hz and 63-80 Hz. In addition, due to the propagation of low-frequency surface waves around the tree grid toward distant observation points, the uncertainty in attenuation performance at those points tends to reduce within the low-frequency range while amplifying across the high-frequency range.  
Z.Z.: conceptualization, formal analysis, investigation, methodology, software, visualization, and writing--original draft; X.Z.: methodology, software, investigation, formal analysis, and writing--review and editing; Z.T.: data curation, formal analysis, methodology, and writing--review and editing; C.H.: data curation, formal analysis, funding acquisition, methodology, and supervision. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the Key Research Project of Shanghai Municipal Commission of Housing and Urban Rural Development grant number 2024-Z02-001-8.  
The raw data supporting the conclusions of this article will be made available by the authors on request.  
The authors declare no conflict of interest.