Abstract: Genomic sequencing revealed that somatic mutations cause a genetic differentiation of the cells in a single tree. In this study, we consider a mathematical model for stem cell proliferation in the shoot apical meristem (abbrev. SAM), which results in genetic diversification between the cells differing in the distance along the shoot and the angle around a shoot axis. The assumptions are as follows. Stem cells in the SAM normally undergo asymmetric cell division and produce successor stem cells and differentiated cells. The differentiated cells proliferate and contribute to shoot elongation. Occasionally, a stem cell is replaced by a copy of an adjacent stem cell. We discuss the "coalescent length" between cells indicating their genetic difference with respect to neutral mutations. A mathematical analysis revealed the following. The genetic diversity of cells sampled at the same position along the shoot increases with the distance from the bottom of the shoot. Stem cells hold a larger variation if they are replaced only by the nearest neighbors than if they are replaced by any cells. The coalescent length between two cells increases not only with the difference in the position along the shoot but also in the angle around the shoot axis. The dynamics of stem cells at the SAM determine the genetic pattern of the entire shoot.