A tree is one rule applied again and again: a branch splits into thinner, shorter branches. The exponent n in r_parent^n = sum r_child^n decides how the sap flows and what the plumbing costs.
Bars: total cross-section area of all branches at a level, divided by the trunk's. Leonardo (n = 2) keeps it at 1.
Relative cost of a symmetric binary tree of the current depth and length ratio, with the trunk radius optimised for each n. Minimum: n = 3.
Builds fresh symmetric trees off-screen and compares the recursion with closed-form results.
The science
The rule. At each fork the radii obey r_p^n = sum r_i^n. With n = 2 the total cross-section area is preserved (Leonardo da Vinci's observation about tree branches; it also follows from the pipe model, where each leaf needs a fixed area of conducting wood). With n = 3 it is Murray's law for vessels. In this model child i gets a share s_i of the parent's r^n, so r_i = r_p s_i^(1/n); a symmetric binary fork gives r_i = r_p 2^(-1/n).
Flow and resistance. Each branch is treated as a pipe: Hagen-Poiseuille resistance R = 8 mu L/(pi r^4). The tree's resistance follows by recursion R(node) = R_seg + 1/sum(1/R(child)) (children in parallel). Every tip transpires equally, so the flow in a branch is Q times (tips below it)/(all tips) and the sap speed is v = Q/(pi r^2). For n = 2 and a symmetric tree the speed is the same in every branch; for n = 3, v is proportional to r, so twigs carry slower sap.
Why 3. The cost of a branch is the pumping power 8 mu L Q^2/(pi r^4) plus upkeep of the wood b pi r^2 L (b is the metabolic cost per unit volume). Minimising over r gives Q proportional to r^3 for every branch, independent of its length, hence Murray's law n = 3 at every fork. The cost chart shows the minimum of the family at n = 3, and the "Murray efficiency" is the cost of the ideal radii divided by the cost of this tree's radii (each with its best overall scale).
Honesty about trees. Real wood conducts through bundles of narrow vessels, not one open pipe, so the absolute resistances here are far too small and are shown only as ratios. Measured plant exponents scatter around 2 to 3 (typical range; trees also have to resist wind and carry their own weight, which favours thicker branches than pure plumbing would). Leonardo's rule, Murray's law and the b-versus-pumping balance are idealised; the shapes (angles, gravitropism, randomness) are only for looks.