Rates of molecular evolution have a central role in our understanding of many aspects of species' biology. However, the causes of variation in rates of molecular evolution remain poorly understood, particularly in plants. Here we show that height accounts for about one-fifth of the among-lineage rate variation in the chloroplast and nuclear genomes of plants. This relationship holds across 138 families of flowering plants, and when accounting for variation in species richness, temperature, ultraviolet radiation, latitude and growth form. Our observations can be explained by a link between height and rates of genome copying in plants, and we propose a mechanistic hypothesis to account for this - the 'rate of mitosis' hypothesis. This hypothesis has the potential to explain many disparate observations about rates of molecular evolution across the tree of life. Our results have implications for understanding the evolutionary history and future of plant lineages in a changing world.