We construct and characterise a 3‐homogeneous but not 2‐primitive permutation groupHof countable degree. It has a transitive extensionJwhich is 5‐homogeneous but not 3‐primitive; the numbernk(J) of orbits ofJonk‐sets is the number of boron trees withkend‐vertices. Some groups related to other classes of trees are also constructed. An application to the growth rate of (nk(G)) for primitive groupsGis given.