Savannas cover one-fifth of the Earth's surface, harbour substantial biodiversity, and provide a broad range of ecosystem services to hundreds of millions of people. The community composition of trees in tropical moist forests varies with climate, but whether the same processes structure communities in disturbance-driven savannas remains relatively unknown. We investigate how biodiversity is structured over large environmental and disturbance gradients in woodlands of eastern and southern Africa. We use tree inventory data from the Socio-Ecological Observatory for Studying African Woodlands (SEOSAW) network, covering 755 ha in a total of 6780 plots across nine countries of eastern and southern Africa, to investigate how alpha, beta, and phylogenetic diversity varies across environmental and disturbance gradients. We find strong climate-richness patterns, with precipitation playing a primary role in determining patterns of tree richness and high turnover across these savannas. Savannas with greater rainfall contain more tree species, suggesting that low water availability places distributional limits on species, creating the observed climate-richness patterns. Both fire and herbivory have minimal effects on tree diversity, despite their role in determining savanna distribution and structure. High turnover of tree species, genera, and families is similar to turnover in seasonally dry tropical forests of the Americas, suggesting this is a feature of semiarid tree floras. The greater richness and phylogenetic diversity of wetter plots shows that broad-scale ecological patterns apply to disturbance-driven savanna systems. High taxonomic turnover suggests that savannas from across the regional rainfall gradient should be protected if we are to maximise the conservation of unique tree communities.