  In recent years the theory of dendroidal sets has emerged as an important
framework for higher algebra. In this article we introduce the concept of a
$C^*$-algebraic drawing of a dendroidal set. It depicts a dendroidal set as an
object in the category of presheaves on $C^*$-algebras. We show that the
construction is functorial and, in fact, it is the left adjoint of a Quillen
adjunction between combinatorial model categories. We use this construction to
produce a bridge between the two prominent paradigms of noncommutative geometry
via adjunctions of presentable $\infty$-categories, which is the primary
motivation behind this article. As a consequence we obtain a single mechanism
to construct bivariant homology theories in both paradigms. We propose a
(conjectural) roadmap to harmonize algebraic and analytic (or topological)
bivariant K-theory. Finally, a method to analyse graph algebras in terms of
trees is sketched.
