## Introduction

Marking, counting and measuring individuals and periodically re-counting these individuals to infer population changes is common practice for ecological studies (Begon *et al*. 1996; Krebs 1999). For studies of tropical trees this has mostly taken the form of census-based permanent plot data: delineating an area, and permanently and uniquely marking each tree stem within that area that exceeds a given lower-size threshold, often ≥ 10 cm d.b.h. (diameter at breast height, 1.3 m, or above buttresses or other bole deformities). The plot is revisited periodically to note deaths and add the trees that reach the lower size limit, and these data are then used to calculate mortality and recruitment estimates (Sheil *et al*. 1995). Almost everything we know about tropical forest dynamics derives from this method (e.g. Connell *et al*. 1984; Swaine *et al*. 1987a; Carey *et al*. 1994; Phillips & Gentry 1994; Condit *et al*. 1999; Sheil *et al*. 2000; Vásquez & Phillips 2000; Lewis *et al*. 2004a; Phillips *et al*. 2004).

Mortality and recruitment estimates are fundamental descriptors of tropical forest tree populations. Comparisons both between and among studies are important if we are to further understand tropical forest dynamics, both to make generalizations about patterns in time and space, and to infer their underlying causes (Swaine *et al*. 1987b; Hartshorn 1990; Phillips & Gentry 1994; Phillips *et al*. 1994; Phillips 1996; Lewis *et al*. 2004a; Phillips *et al*. 2004). However, these recruitment and mortality rate estimates are affected by a potentially serious artefact that may influence any conclusion based on permanent sample plots: these rate estimates are not independent of the time interval between censuses (Sheil & May 1996). Thus, conclusions based on comparing rates with differing census intervals are open to debate (for example see Phillips & Gentry 1994; responses by Phillips 1995; Sheil 1995; Phillips *et al*. 2004).

Mortality rate estimates are often based on models that assume a population is homogenous, with each member having an equal and constant probability of dying (Sheil *et al*. 1995). Sheil and May (1996) show theoretically that when a population is made up of subpopulations with differing mortality rates, the population mortality rate, on average, will decrease with increasing time between censuses. This is because higher-mortality stems die faster, leaving increasing proportions of the original cohort represented by lower-mortality stems. Over time the lower-mortality stems dominate, leading to lower estimates of population mortality rates as the census interval increases.

It is unknown whether the theoretically demonstrated decline in mortality rate with increasing census interval is a serious problem or essentially trivial over the time-scales on which ecologists measure forest dynamics. Certainly, tropical forest stands are not homogeneous with respect to the mortality, recruitment or turnover of their component subpopulations. For example, individual species’ mortality rates in three different forest stands each appear to vary by over an order of magnitude (Vanclay 1991; Favrichon 1994; Condit *et al*. 1995). Further within-stand heterogeneity may be generated if mortality rates change with stem size, as has been shown for stems ≥ 10 cm d.b.h. in several localities (Mervart 1972; Hartshorn 1990; Hubbell & Foster 1990; Vanclay 1991; Clark & Clark 1992). Additionally, heterogeneity may arise from species-level mortality rates varying according to local variation in the physical environment. Thus we expect some impact of census interval length on mortality rate estimates, but neither *a priori* predictions of the magnitude or form of declines (linear, exponential, etc.), nor initial estimates of the declines in forest stands, are known.

Accurate mortality rate estimation presents several other problems, notably the large sample sizes and long time-periods required, due to intrinsically low mortality rates and the large stochastic component of tree mortality in tropical forests. For example, in a study of 10 ha of forest in Ecuador, 20% of tree deaths > 10 cm d.b.h. were due to other trees knocking them over (Gale & Barfod 1999). Furthermore, very occasional deaths of very large trees accentuate this stochastic element by dramatically altering forest structure and hence local competition and recruitment (Sheil *et al*. 2000). A stand of 1 ha with *c*. 600 trees ≥ 10 cm d.b.h. and a mortality rate of *c*. 1.5% a^{−1} translates to *c*. 9 trees dying ha^{−1} a^{−1}, hence stochastic effects altering this by only a few trees can have large effects on average mortality rates. Thus long-term and/or large-scale studies are required.

Studies of tropical tree dynamics often consider mortality but not recruitment (Swaine *et al*. 1987b; Hartshorn 1990; Vanclay 1991; Condit *et al*. 1995; Lieberman *et al*. 1996); however, many studies report both (Carey *et al*. 1994; Favrichon 1994; Phillips & Gentry 1994; Phillips 1996; Condit *et al*. 1999; Sheil *et al*. 2000). Sheil and May (1996) show that the same effect of census interval applies to recruitment rates: a decline with increasing census interval. Of course, if recruitment rate estimates chosen are calculated as the number of recruited stems needed to maintain a population in equilibrium, as is commonly the case, then, by definition, declines with increasing census length are predicted to match those for mortality. Stem turnover (*sensu*Phillips & Gentry 1994), the mean of the mortality and recruitment rates of a forest stand, would also, by definition, show the same decline with increasing census interval.

As more studies on forest dynamics are completed the scope for comparative studies and syntheses of tropical forest ecology increases, so there is an urgent need to understand and quantify the potential census-interval artefact. For example, one study of 65 sites from across the tropics showed that turnover rates have been increasing over the late 20th century, possibly as a result of increased productivity (Phillips & Gentry 1994; see Phillips 1996). However, the study included sites with a wide range of census intervals (2–38 years), and the central finding has hence been controversial (Phillips 1995; Sheil 1995; Sheil & May 1996; Condit 1997; Phillips & Sheil 1997; Phillips *et al*. 2004).

We take two approaches to assess if the census interval effect on tropical forest tree mortality, recruitment and turnover rates is a serious problem. First, we use species-level mortality rates from seven sites from Latin America, Africa, Asia and Australia, using each species as a subpopulation to calculate the expected mortality rate for each stand for census intervals from 1 up to 50 years. This gives a first estimate of the magnitude of the artefact, and a clue as to the shape of the relationship between mortality rates and census interval length. Secondly, we use 14 long-term multicensus plots from Latin America, Africa, Asia and Australia to test for a decline in mortality over time in permanent sample plot data. This gives a second estimate of the decline and its form, but will be more variable as stochastic effects and changes in mortality over time are included in the long-term data. Finally, we re-analyse Phillips's (1996) data set of 65 sites to test his conclusions about changes in forest dynamics over time and regional differences in dynamics, taking into account the differing census intervals used in each of the 65 sites.