Taylor's law says that the variance of population density of a species is proportional to a power of mean population density. Density–mass allometry says that mean population density is proportional to a power of mean biomass per individual. These power laws predict a third, variance–mass allometry: the variance of population density of a species is proportional to a power of mean biomass per individual. We tested these laws using 10 censuses of New Zealand mountain beech trees in 250 plots over 30 years at spatial scales from 5 m to kilometers. We found that: 1) a single-species forest not disrupted by humans obeyed all three laws; 2) random sampling explained the parameters of Taylor's law at a large spatial scale in 8 of 10 censuses, but not at a fine spatial scale; 3) larger spatial scale increased the exponent of Taylor's law and decreased the exponent of variance–mass allometry (this is the first empirical demonstration that the latter exponent depends on spatial scale), but affected the exponent of density–mass allometry slightly; 4) despite varying natural disturbance, the three laws varied relatively little over the 30 years; 5) self-thinning and recruiting plots had significantly different intercepts and slopes of density–mass allometry and variance–mass allometry, but the parameters of Taylor's law were not usually significantly affected; and 6) higher soil calcium was associated with higher variance of population density in all censuses but not with a difference in the exponent of Taylor's law, while elevation above sea level and soil carbon-to-nitrogen ratios had little effect on the parameters of Taylor's law. In general, the three laws were remarkably robust. When their parameters were influenced by spatial scale and environmental factors, the parameters could not be species-specific indicators. We suggest biological mechanisms that may explain some of these findings.