  For every $n\in \mathbb{N}$, we present a set $S_n$ of $O(n^{3/2}\log n)$
points in the plane such that every planar 3-tree with $n$ vertices has a
straight-line embedding in the plane in which the vertices are mapped to a
subset of $S_n$. This is the first subquadratic upper bound on the size of
universal point sets for planar 3-trees, as well as for the class of 2-trees
and serial parallel graphs.
