In principle, our way of finding the optimal diameter was wrong: Murray’s law is satisfied by optimal systems, but is not a recipe to find the optimum. To do so, one must know the cost parameters (relation between the two terms and relative ‘importance’ of each branch), and then calculate the cost. When we just fix two branches and calculate the third with Murray’s law, we are implicitly changing the parameters of the cost (we go to a bifurcation whose ‘best fitting’ parameters are different than the initial ones). If we calculate the new cost with the old parameters, in some cases it may increase instead of decrease. For example:

>> d1=2; d2=1; d3=1;

>> C=L1*d1^2 + L1/d1 + L2*d2^2 + .5*L2/d2 + L3*d3^2 + .5*L3/d3

C =

12.7500

>> d1^3-d2^3-d3^3

ans =

6

>> d2=(d1^3-d3^3)^(1/3);

>> d1^3-d2^3-d3^3

ans =

1.7764e-015

>> C=L1*d1^2 + L1/d1 + L2*d2^2 + .5*L2/d2 + L3*d3^2 + .5*L3/d3

C =

16.3810

However, I think it is possible to do it correctly. For example, one can first calculate reasonable parameters for the network, and then calculate the optimum for each branch, with respect to those parameters. Also, do not think that the results found so far are completely meaningless: Although they are strictly incorrect, they do indicate a trend to larger deviations for shorter branch lengths.