  In the longest plane spanning tree problem, we are given a finite planar
point set $\mathcal{P}$, and our task is to find a plane (i.e., noncrossing)
spanning tree for $\mathcal{P}$ with maximum total Euclidean edge length.
Despite more than two decades of research, it remains open whether this problem
is NP-hard. Thus, previous efforts have focused on olynomial-time algorithms
that produce plane trees whose total edge length approximates $\text{OPT}$, the
maximum possible length. The approximate trees in these algorithms all have
small unweighted diameter, typically three or four. It is natural to ask
whether this is a common feature of longest plane spanning trees, or an
artifact of the specific approximation algorithms.
  We provide three results to elucidate the interplay between the approximation
guarantee and the unweighted diameter of the approximate trees. First, we
describe a polynomial-time algorithm to construct a plane tree with diameter at
most four and total edge length at least $0.546 \cdot \text{OPT}$. This
constitutes a substantial improvement over the state of the art. Second, we
show that a longest plane tree among those with diameter at most three can be
found in polynomial time. Third, for any candidate diameter $d \geq 3$, we
provide upper bounds on the approximation factor that can be achieved by a
longest plane tree with diameter at most $d$ (compared to a longest plane tree
without constraints).
