I will present some of the combinatorics that came up in my work on configuration spaces of points. Trees arise in the Fulton-MacPherson compactification X[n] as a set indexing its strata, and the collection {X[n]}, for n positive, forms a module over the operad of `local' configurations. A direct connection between X[n] and the associahedra is known too. In a related compactification that I studied, trees acquire some extra structure, levels of vertices, while the permutahedra replace the associahedra. There is a clear iterated compositional structure, which is not operadic though.