  We prove that all knots with unknotting number at most 21 are smoothly slice
in the K3 surface. We also prove a more general statement for 4-manifolds that
contain a plumbing tree of spheres. Our strategy is based on a flexible method
to remove double points of immersed surfaces in 4-manifolds by tubing over
neighbourhoods of embedded trees. As a byproduct, we recover a classical result
of Norman and Suzuki that every knot is smoothly slice in $S^2 \times S^2$ and
in $\mathbb{CP}^2 \# \overline{\mathbb{CP}^2}$.
