  The central aim of this work is to understand rough differential equations on
homogeneous spaces. We focus on the formal approach, by giving an explicit
expansion of the solution at each point of the real line in terms of decorated
planar forests. For this we develop the notion of planarly branched rough
paths, following M. Gubinelli's branched rough paths. The definition is similar
to the one in the flat case, the main difference being the replacement of the
Butcher--Connes--Kreimer Hopf algebra of non-planar rooted forests by the
Munthe-Kaas--Wright Hopf algebra of planar rooted forests. We show how the
latter permits to handle rough differential equations on homogeneous spaces
using planarly branched rough paths, the same way branched rough paths are used
in the context of rough differential equations on finite-dimensional vector
spaces. An analogue of T. Lyons' extension theorem is proven. Finally, under
analyticity assumptions on the coefficients and when the H\"older index of the
driving path is equal to one, we show convergence of the planar forest
expansion in a small time interval.
