Measured values of four key leaf traits (leaf area per unit mass, nitrogen concentration, photosynthetic capacity, leaf lifespan) co-vary consistently within and among diverse biomes, suggesting convergent evolution across species. The same leaf traits co-vary consistently with the environmental conditions (light intensity, carbon-dioxide concentration, nitrogen supply) prevailing during leaf development. No existing theory satisfactorily explains all of these trends. Here, using a simple model of the carbon-nitrogen economy of trees, we show that global leaf-trait relationships and leaf responses to environmental conditions can be explained by the optimization hypothesis (MAXX) that plants maximize the total amount of carbon exported from their canopies over the lifespan of leaves. Incorporating MAXX into larger-scale vegetation models may improve their consistency with global leaf-trait relationships, and enhance their ability to predict how global terrestrial productivity and carbon sequestration respond to environmental change.