As plants grow, crowding eventually occurs and competition leads to size- and density-dependent mortality. Close packing of plants is maintained throughout, such that the growth mode of individual plants strictly dictates the exponent $\alpha$ in the power function V = kN$^\alpha$, relating average plant volume, V, to population density, N. Plant characteristics define the self-thinning line in a log-log diagram as follows. The growth mode determines the way in which plant form and structure change with growth, thereby dictating the slope $\alpha$. The specific form and structure of plants, and the overlap between neighbors, determine the local elevation of the thinning curve but have no direct effect on $\alpha$. The specific form and structure, crowding, and the growth mode jointly determine the y-intercept k. Various growth modes are examined and the concomitant thinning functions derived. The geometric-similarity model makes remarkably accurate predictions of the thinning exponent -1.5 obtained empirically from populations of mosses, ferns, grasses, herbs, and some trees. But the -3/2 slope may be mimicked by other growth modes. The growth geometry of most trees generally conforms much better to the elastic-similarity principle. A model based on elastic similarity predicts a thinning curve with gradients ranging from -2 to -1.33, with the most realistic range being -1.97 to -1.80. Populations of many tree species exhibit thinning gradients in accord with the elastic-similarity model (and empirical allometry), but others come closer to the geometric-similarity exponent, -1.5. The perplexing incompatibility between allometry and thinning gradients in some trees might be a result of root competition, in combination with different growth modes for roots and aerial parts, with the roots dictating the thinning gradient of aboveground structures. Another model, based on a stagnation of the height growth of trunks or of the depth growth of roots, may explain the thinning gradient -1 for trunks of big, old trees. When root competition drives population thinning, the thinning gradient could easily be at variance with the empirical allometry of aboveground structures, and progressively more open space among plants may open up as they grow, with crowding being restricted to the roots. This may explain the wide spacings usually seen between bushes and trees growing on shallow soils where depth may be limited by bedrock or by a high water table (as in bogs); the thinning gradient -1 should then prevail. In most trees, the leaves remain about the same size throughout the growth of the tree. This constancy of leaf size might be expected to result in radically different thinning slopes for leaves than for trunk and branches. But I show that as long as the canopy layer still grows in depth, the thinning exponent for leaf volume per tree versus density of the tree population may be -1.5, regardless of whether leaves remain of a constant size or grow isometrically with respect to trunk and branches. After the foliage layer has ceased to grow in thickness, the thinning exponent should be -1.5 for leaves that grow isometrically with stem and branches, but -1 for leaves of constant size. Gradients in thinning functions reflect size-related design principles that apply regardless of whether the plants are different species in different populations (in across-species regressions) or the same individuals at different growth stages, in which case the gradient reflects the growth mode of individual plants. The predicted thinning gradients for various growth modes are all readily testable. The adherence to a common growth mode, or structural-design principle, explains why different plant species exhibit the same thinning gradient, despite profound dissimilarities in structure and geometry. If geometric similarity applies throughout the smaller size classes of plants and animals (as seems to be the rule), then the safety factor against buckling becomes progressively narrower with increasing weight. Selection for the elastic-similarity principle will therefore be greater among large plants (like trees) and animals than among smaller organisms.