Aggregation patterns are often visually detected in sets of location data. These clustersmay be the result of interesting dynamics or the effect of pure randomness. We build anasymptotically Gaussian test for the hypothesis of randomness corresponding to ahomogeneous Poisson point process. We first compute the exact first and second moment ofthe Ripley K-statistic under the homogeneous Poisson point process model.Then we prove the asymptotic normality of a vector of such statistics for different scalesand compute its covariance matrix. From these results, we derive a test statistic that ischi-square distributed. By a Monte-Carlo study, we check that the test is numericallytractable even for large data sets and also correct when only a hundred of points areobserved.

Spatial analysis can be used to relate the patterns of tree species to their regeneration syndromes – pioneer to late-successional – and is a first step in refining hypotheses about the species traits and biotic and abiotic factors that give rise to forest community dynamics. This study examines the spatial pattern of the most abundant trees in three 0.45-ha plots in species-poor lowland rain forests on oceanic islands in Tonga, Western Polynesia, that experience frequent natural disturbance and have a 3000-y history of shifting cultivation. We contrast secondary vs. remnant late-successional forest, with particular attention paid to the spatial dispersion and clustering of tree species, and the presence of spatial dependence in the density of seedlings and saplings. Shade-tolerant species were not strongly clustered at any scale. They did not appear to be dispersal limited, in late successional forest, and only some showed patterns consistent with density-dependent mortality (more clumped when small). Shade-tolerant species were more clumped in secondary forest, and may be dispersal-limited there because vertebrate dispersers prefer primary forest. Shade-intolerant species were clumped in gaps in late-successional forest, but some were also clumped in secondary forest, indicating that they too may be dispersal limited during secondary succession. We also compared the species composition of seedlings and saplings in the centre of plots with trees in the surrounding area and inferred that active dispersal (by vertebrate frugivores) contributed as much as 50% to site species richness.