The simplex category may be regarded as the category of all linear directed graphs. The tree category generalizes this to directed rooted trees.
Finite planar level tree
(… ) see Berger
Finite symmetric rooted trees
We define the category finite symmetric rooted trees.
The objects of are non-empty non-planar trees with specified root.
Each such tree may naturally be regarded as specifying an (colored) symmetric operad with one color per edge of the tree. A morphism of trees in is a morphism of the corresponding operads.
As such, is by construction a full subcategory of that of symmetric operads enriched over Set.
A presheaf on is a dendroidal set, a generalization of a simplicial set.