The quantification of tropical tree biodiversity worldwide remains an open and challenging problem. In fact, more than two-fifths of the global tree population can be found either in tropical or sub-tropical forests, but species identities are known only for $\approx$0.000067$\%$ of the individuals in all tropical forests. For practical reasons, biodiversity is typically measured or monitored at fine spatial scales. However, important drivers of ecological change tend to act at large scales. Conservation issues, for example, apply to diversity at global, national or regional scales. Extrapolating species richness from the local to the global scale is not straightforward. Indeed, a vast number of different biodiversity estimators have been developed under different statistical sampling frameworks, but most of them have been designed for local/regional-scale extrapolations, and they tend to be sensitive to the spatial distribution of trees, sample coverage and sampling methods. Here, we introduce an analytical framework that provides robust and accurate estimates of species richness and abundances in biodiversity-rich ecosystems, as confirmed by tests performed on various in silico-generated forests. The new framework quantifies the minimum percentage cover that should be sampled to achieve a given average confidence in the upscaled estimate of biodiversity. Our analysis of 15 empirical forest plots shows that previous methods have systematically overestimated the total number of species and leads to new estimates of hyper-rarity at the global scale, known as Fisher’s paradox. We show that hyper-rarity is a signature of critical-like behavior in tropical forests, and it provides a buffer against mass extinctions. When biotic factors or environmental conditions change, some of these rare species are more able than others to maintain the ecosystem’s functions, thus underscoring the importance of rare species.