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

Appendix B. Greenhill scaling and fractal dimension in the branching network.

In the branching network, the length, L_k, from any level k to the branch tip is

(B.1)

where l_i is internodal length at level i. Because the sum of the infinite series is Because scales as this also means that
. If = 2/3, the fractal trees follow Greenhill scaling, which accords with the assumption of elastic similarity (McMahon 1973).

Assume that the fractal dimension of the foliage is z. Then, by definition, where is the foliage attached to the branch tips generated by the internode at level k. The total amount of foliage in the branching network is hence

(B.2)

where is the number of internodes at level k and n is the number of ramifications at each internode. Because , we require that which obtains if and only if

(B.3)