Grow a Tree
A cloud of light fills the crown and every tip grows toward the light it can see, using it up as it arrives. The shape is a negotiation, not a plan. The trunk is never told how thick to be: at every fork the cross-section of the branches adds up to the branch they came from, and the page measures that ratio live — exactly 1 at Leonardo's exponent of 2.
Nothing in this tree is drawn on purpose. There is no list of branches and no angle that says where the next one goes. There is a seed at the bottom of the picture and a cloud of points of light where the crown should be, and one rule: every point pulls on the nearest growing tip it can see, every tip grows one step toward the average of the points pulling on it, and a point that a tip reaches is used up. Run that a few dozen times and a tree comes out — a trunk that climbs until it can see the light, then forks and forks again as the light around it divides.
The surprise is how much of a tree's character lives in the shape of its light rather than in its wood. Fill an oval with light and you get a broad cherry; a cone gives a fir; light hung in curtains with a little gravity on every step gives a willow; light blown into a streak by a coastal wind gives a pine leaning into it. Branches do not cross, because two tips never compete for the same point for long — the first to arrive consumes it. Paint new light in an empty part of the sky and a branch leaves the crown to go and get it. Cut a branch and the light where it stood is free again, so the tree grows back into the gap.
The thickness comes from somewhere much older. Leonardo da Vinci noted in his notebooks that all the branches of a tree at every stage of its height, put together, are equal in thickness to the trunk. Read as cross-sectional area, that is the pipe model: if each twig is a bundle of the same pipes, a fork just splits the bundle, and the radius of a parent is the square root of the sum of its children's squared radii. Here every twig gets the same radius and the rule is applied from the tips down, so the trunk thickens as the crown fills without ever being told to. The exponent is a slider: 2 is Leonardo's, and measurements of real trees mostly fall close to it.
The same idea — a branching network that grows toward what it needs and stops where it is satisfied — has been used to model leaf veins, blood vessels, river networks and the roots of plants searching for water, and to grow the trees in films and games without anyone placing a single branch. It is a good example of a shape that is better described by the process that made it than by any drawing of it.
A car behind one door, goats behind two. The host opens a goat. Thousands of readers, PhDs among them, swore switching makes no difference — play ten thousand games and watch them lose the argument.