  A queue layout of a graph G consists of a linear order of the vertices of G
and a partition of the edges of G into queues, so that no two independent edges
of the same queue are nested. The queue number of G is the minimum number of
queues required by any queue layout of G.
  In this paper, we continue the study of the queue number of planar 3-trees.
As opposed to general planar graphs, whose queue number is not known to be
bounded by a constant, the queue number of planar 3-trees has been shown to be
at most seven. In this work, we improve the upper bound to five. We also show
that there exist planar 3-trees, whose queue number is at least four; this is
the first example of a planar graph with queue number greater than three.
