Anniki Mäkelä and Harry T. Valentine. 2006. Crown ratio influences allometric scaling in trees. Ecology 87:2967–2972.

Appendix D. Scaling in the whole tree.

The assumption that A = r_CB allows us to estimate the volume of the bare trunk, V_D, with the trapezoidal rule, i.e.,

(D.1)

where B and A are the cross-sectional areas at the base of the trunk and the base of the crown, H is tree height and L₀ is crown length, so H-L₀ is trunk length.

Eq. C.1 implies that the woody volume in the crown, is proportional to AL₀. To compare this with the volume of the bare trunk we define as a constant of proportionality, such that Here, is the average pipe length of active and disused woody pipes (Valentine 1985). Total woody volume is hence

(D.2)

implying that where M_T denotes total woody mass. In other words, foliage scales as but the level of this scaling relationship depends on crown ratio. Exactly how crown ratio affects the scaling relationship depends on the average pipe length, especially the constant .

The average pipe length can be determined by evaluating the infinite sum in Eq. C.1, which yields

(D.3)

Obviously, mean pipe length equals L₀ if a = 1, and reduces with increasing a. For example, for n = 2, a=1.5 and z = 2.5, ½ = 0.52; for n=3, ½ = 0.56. Empirical evidence suggests that average pipe length is approximately 0.5L₀ ( 1 ) in several species (Kantola and Mäkelä 2006, Valentine and Mäkelä 2005, Ilomäki et al. 2003). In any case, ( - 1 ) is small compared with r_C^-2, implying that M r_C²M_T (Fig. A1). This shows that Eq. 3 holds approximately.

FIG. D1. Relationship between total above-ground woody mass (MT) and woody mass in the crown (M). rC is crown ratio. Circles and solid black line: pine (Pinus sylvestris); triangles and dashed line with dots: spruce (Picea abies); squares and long dashed line: birch (Betula pendula). The lines are linear least-squares fits with zero intercept to the data. Pine: y = 0.965x, R2 = 0.9366, N=28; spruce: y = 1.1414x, R2 = 0.9826, N =2 9; birch: y = 1.0085x, R2 = 0.9997, N = 18.

LITERATURE CITED

Ilomäki, S., E. Nikinmaa, and A. Mäkelä. 2003. Crown rise due to competition drives biomass allocation in silver birch. Canadian Journal of Forest Research 33:2395–2404.

Kantola, A., and A. Mäkelä. 2006. Development of biomass proportions in Norway spruce (Picea abies [L.] Karst.). Trees (in press).

Valentine, H. T., and A. Mäkelä. 2005. Bridging process-based and empirical approaches to modeling tree growth. Tree Physiology 25:769–779.