Structure tree

Definition with symbols
Let $$G$$ be a group acting transitively on a set $$S$$. A structure tree for the action of $$G$$ on $$S$$ is a tree described as follows:


 * Put all the elements of $$S$$ at the leaves of the tree
 * Find a minimal nontrivial block decomposition for $$S$$, or equivalently, a decomposition into blocks such that each block contains no proper nontrivial block. Represent each block by a point, and make this set of points form the layer of the tree just above the leaves, such that the children of each block are the elements of that block.
 * Now, $$G$$ acts on the set of blocks. Treat this as the new set, again find a minaml block decomposition and make the next layer.
 * Keep repeating this process till the action of $$G$$ on the blocks becomes primitive, in which case the next layer will be the root (The block representing the whole of $$G$$)

Example
(I will also put a diagram to make things clearer).