. 2009 Apr 10;106(17):7040–7045. doi: 10.1073/pnas.0812294106

Table 1.

Similarity of predicted scaling relations for branches within a tree [quantities denoted by uppercase symbols and subscripts i (46)], and for trees within a forest (denoted by lowercase symbols and subscripts k)^*

Scaling quantity	Individual tree	Entire forest
Area preserving	$\frac{R_{i} + 1}{R_{i}} = \frac{1}{n^{1 / 2}}$	$\frac{r_{k} + 1}{r_{k}} = \frac{1}{λ^{1 / 2}}$
Space filling	$\frac{L_{i} + 1}{L_{i}} = \frac{1}{n^{1 / 3}}$	$\frac{l_{k} + 1}{l_{k}} = \frac{1}{λ^{1 / 3}}$
Biomechanics	R_i² = L_i³	r_k² = l_k³
Size distribution^*	ΔN_i ∝ R_i⁻² ∝ M_i^−3/4	Δn_k ∝ r_k⁻² ∝ m_k^−3/4
Energy and material flux^*	B_i ∝ R_i² ∝ N_i^L ∝ M_i^3/4	B_k ∝ r_k² ∝ n_k^L ∝ m_k^3/4

*The above theory is developed based on using radius as the primary measure of size. The dependences on mass, leading to quarter-power exponents, are derived expressions using the continuous distribution function f(r) ∝ 1/r² (Eq. 8 and SI Text, Eq. S3). The mathematical equivalence of these scaling relations shows that the entire forest behaves as if it were a hierarchically branching resource supply network that mimics the branching network of a single tree (see also Fig. 3).