The phyllosphere--the aerial surfaces of plants, including leaves--is a ubiquitous global habitat that harbors diverse bacterial communities. Phyllosphere bacterial communities have the potential to influence plant biogeography and ecosystem function through their influence on the fitness and function of their hosts, but the host attributes that drive community assembly in the phyllosphere are poorly understood. In this study we used high-throughput sequencing to quantify bacterial community structure on the leaves of 57 tree species in a neotropical forest in Panama. We tested for relationships between bacterial communities on tree leaves and the functional traits, taxonomy, and phylogeny of their plant hosts. Bacterial communities on tropical tree leaves were diverse; leaves from individual trees were host to more than 400 bacterial taxa. Bacterial communities in the phyllosphere were dominated by a core microbiome of taxa including Actinobacteria, Alpha-, Beta-, and Gammaproteobacteria, and Sphingobacteria. Host attributes including plant taxonomic identity, phylogeny, growth and mortality rates, wood density, leaf mass per area, and leaf nitrogen and phosphorous concentrations were correlated with bacterial community structure on leaves. The relative abundances of several bacterial taxa were correlated with suites of host plant traits related to major axes of plant trait variation, including the leaf economics spectrum and the wood density-growth/mortality tradeoff. These correlations between phyllosphere bacterial diversity and host growth, mortality, and function suggest that incorporating information on plant-microbe associations will improve our ability to understand plant functional biogeography and the drivers of variation in plant and ecosystem function.