Higher-rank trees: Finite higher-rank planar trees arising from polyhedral graphs. Differences from one-trees

We introduce a new family of higher-rank graphs, whose construction was inspired by the graphical techniques of Lambek {Lambek} and Johnstone {Johnstone} used for monoid and category emedding results. We show that they are planar k-trees for 2 ≤ k ≤ 4. We also show that higher-rank trees differ from $1$-trees by giving examples of higher-rank trees having properties which are impossible for $1$-trees. Finally, we collect more examples of higher-rank planar trees which are not in our family.