Particles that stick less often slip past the tips: thicker, denser branches.
A pull toward the cluster: straighter, faster branches.
How strongly the field picks the next step: low, a bushy discharge; high, a single bolt.
Click the picture to plant a new seed (not for lightning). The address of the page keeps the settings, to share them.
A random walk that builds
In 1981 the physicists Thomas Witten and Leonard Sander proposed a very simple model. Release a particle far away and let it wander, one random step after another. When it touches the cluster, it sticks. Release the next one. They called it diffusion-limited aggregation, and the clusters it grows are fractals: branches on branches, never filling in.
The reason is that tips stick out. A wandering particle is much more likely to bump into a tip than to find its way down into a fjord between two branches, so tips grow faster, and the faster they grow, the more they stick out. The same runaway shapes copper deposits in an electrolyte, mineral dendrites in rocks, frost on a window, coral, river networks, and the branching path of a lightning bolt, whose leader grows where the electric field is strongest, at its tips. For lightning, this page uses the dielectric breakdown model of Niemeyer, Pietronero and Wiesmann (1984), the electric version of the same idea: the bolt steps down to wherever the field pulls hardest, and when a branch touches the ground, the path it opened lights up.
Stickiness lets particles slip past before they stick, so they reach further in and the branches thicken. Drift pulls them toward the cluster, like gravity on coral polyps or the field of a storm cloud, and straightens the branches.