The simplex category may be regarded as the category of all linear directed graphs. The tree category generalizes this to directed rooted trees.
(… ) see Berger
We define the category $\Omega$ finite symmetric rooted trees.
The objects of $\Omega$ 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 $\Omega$ is a morphism of the corresponding operads.
As such, $\Omega$ is by construction a full subcategory of that of symmetric operads enriched over Set.
A presheaf on $\Omega$ is a dendroidal set, a generalization of a simplicial set.
Clemens Berger, A cellular nerve for higher categories, Advances in Mathematics 169, 118-175 (2002) (pdf)
See the references at dendroidal set.