whatA leaf blade sprinkles auxin sources -- spots calling for plumbing. Every vein tip grows toward the calls nearest it; an answered call goes quiet; and when two advancing veins meet, they JOIN, closing a loop instead of keeping the family tree. Sap is then solved as current in a resistor network, and thickness follows flow.

whyThe claim: loops, not trees, and the loops are the point. A pure tree is efficient and fragile -- sever one rib and everything beyond it starves. Click any vein to cut it and watch the sap visibly reroute, thicknesses reshuffling as flow finds its way around the closed circuits: measured here, better than nine tenths of the leaf still drains after losing its busiest vein. Real leaves anastomose because insects bite them.

howCanalization in the botanists' style with a joining rule for anastomosis; growth consumes sources and resprinkles until the blade is served. The flow solve relaxes a Laplacian with unit sap at every tip and the petiole grounded. Cuts heal after a few seconds; left alone the leaf wounds itself now and then to show off. One integer seeds the blade and every vein.