1 Introduction

2 Post-Lie algebras and the Guin–Oudom construction

3 Munthe-Kaas–Wright Hopf algebra of planar forests

4 Hopf algebras with a natural growth operator

5 Natural growth in the MKW Hopf algebra

6 Rough paths in a combinatorial Hopf algebras

7 Planarly branched rough paths are geometric

8 Geometric embedding for planar regularity structures

9 Conclusions