When Darwin visited the Galapagos archipelago, he observed that, in spite of the islands' physical similarity, members of species that had dispersed to them recently were beginning to diverge from each other. He postulated that these divergences must have resulted primarily from interactions with sets of other species that had also diverged across these otherwise similar islands. By extrapolation, if Darwin is correct, such complex interactions must be driving species divergences across all ecosystems. However, many current general ecological theories that predict observed distributions of species in ecosystems do not take the details of between-species interactions into account. Here we quantify, in sixteen forest diversity plots (FDPs) worldwide, highly significant negative density-dependent (NDD) components of both conspecific and heterospecific between-tree interactions that affect the trees' distributions, growth, recruitment, and mortality. These interactions decline smoothly in significance with increasing physical distance between trees. They also tend to decline in significance with increasing phylogenetic distance between the trees, but each FDP exhibits its own unique pattern of exceptions to this overall decline. Unique patterns of between-species interactions in ecosystems, of the general type that Darwin postulated, are likely to have contributed to the exceptions. We test the power of our null-model method by using a deliberately modified data set, and show that the method easily identifies the modifications. We examine how some of the exceptions, at the Wind River (USA) FDP, reveal new details of a known allelopathic effect of one of the Wind River gymnosperm species. Finally, we explore how similar analyses can be used to investigate details of many types of interactions in these complex ecosystems, and can provide clues to the evolution of these interactions.