We study unital \(\infty \) -operads by their arity restrictions. Given \(k \ge 1\) , we develop a model for unital k-restricted \(\infty \) -operads, which are variants of \(\infty \) -operads which have only \((\le k)\) -arity morphisms, as complete Segal presheaves on closed k-dendroidal trees, which are closed trees built from corollas with valences \(\le k\) . Furthermore, we prove that the restriction functors from unital \(\infty \) -operads to unital k-restricted \(\infty \) -operads admit fully faithful left and right adjoints by showing that the left and right Kan extensions preserve complete Segal objects. Varying k, the left and right adjoints give a filtration and a co-filtration for any unital \(\infty \) -operad by k-restricted \(\infty \) -operads.
