The Amazon rain forest sustains the world's highest tree diversity, but it remains unclear why some clades of trees are hyperdiverse, whereas others are not. Using dated phylogenies, estimates of current species richness and trait and demographic data from a large network of forest plots, we show that fast demographic traits – short turnover times – are associated with high diversification rates across 51 clades of canopy trees. This relationship is robust to assuming that diversification rates are either constant or decline over time, and occurs in a wide range of Neotropical tree lineages. This finding reveals the crucial role of intrinsic, ecological variation among clades for understanding the origin of the remarkable diversity of Amazonian trees and forests.