The pictures below show an example, based on the rule 184 cellular automaton. Starting from random initial conditions this rule yields a collection of stripes which annihilate whenever they meet, leading to a sequence of progressively larger nested regions.
And as the pictures show, these regions form a pattern that corresponds to a random tree that builds up from its smallest branches, much in the way that a river builds up from its tributaries.
Nesting in rule 184 is easiest to see when the initial conditions contain exactly equal numbers of black and white cells, so that the numbers of left and right stripes exactly balance, and all stripes eventually annihilate. But even when the initial conditions are such that some stripes survive, nested regions are still formed by the stripes that do annihilate. And indeed in essentially any system where there are domains that grow fairly independently and then progressively merge the same basic overall nesting will be seen.
As an example, the picture below shows the rule 110 cellular automaton evolving from random initial conditions. The picture