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• Equations were based on rates of change rather than state variables. • Self-thinning equations accounted for species mixing, dominance, site, and climate. • All-species equations were also developed, and could be used for rarer species. • Based on temperate, Mediterranean and subtropical stands in Australia, China and Switzerland (1579 plots). Self-thinning dynamics are often considered when managing stand density in forests and are used to constrain forest growth models. However, self-thinning relationships are often quantified using only data at a conceptualised self-thinning line, even though self-thinning can begin before the stand actually reaches a self-thinning line. Also, few self-thinning relationships account for the effects of species composition in mixed-species forests, and stand structure such as relative height of species (in mixtures), and/or size or age cohorts in uneven-aged forests. Such considerations may be important given the effects of global climate change and interest in mixed-species and uneven-aged forests. The objective of this study was to develop self-thinning relationships based on changes in the tree density relative to mean tree diameter, instead of focusing only on data for state variables (e.g. tree density) at the self-thinning line. This was done while also considering how the change in tree density is influenced by site quality and stand structure (species composition and relative height). The relationships were modelled using data from temperate Australian Eucalyptus plantations (436 plots), subtropical forests in China (88 plots), and temperate forests in Switzerland (1055 plots). Zero-inflated and hurdle generalized linear models with Poisson and negative binomial distributions were fit for several species, as well as for all-species equations. The intercepts and slopes of the self-thinning lines were higher than many published studies which may have resulted from both the less restrictive equation form and data selection. The rates of self-thinning often decreased as the proportion of the object species increased, as relative height increased (species or size cohort became more dominant), and as site (quality) index increased. The effects of aridity varied between species, with self-thinning increasing with aridity index for Abies alba, Pinus sylvestris, Quercus petraea and Quercus robur , but decreasing with aridity index for Eucalyptus nitens, Fagus sylvatica and Picea abies as sites became wetter and cooler. Self-thinning model parameters were not correlated with species traits, including specific leaf area, wood basic density or crown diameter – stem diameter allometry. All-species self-thinning relationships based on all data could be adjusted using a correction factor for rarer species where there were insufficient data to develop species-specific equations. The approach and equations developed could be used in forest growth models to calculate how the tree density declines as mean tree size increases, as height changes relative to other cohorts or species, as species proportions change, and as climatic and edaphic conditions change. [ABSTRACT FROM AUTHOR]