  Let $\mathcal{P}$ be a finite set of points in the plane in general position.
For any spanning tree $T$ on $\mathcal{P}$, we denote by $|T|$ the Euclidean
length of $T$. Let $T_{\text{OPT}}$ be a plane (that is, noncrossing) spanning
tree of maximum length for $\mathcal{P}$. It is not known whether such a tree
can be found in polynomial time. Past research has focused on designing
polynomial time approximation algorithms, using low diameter trees. In this
work we initiate a systematic study of the interplay between the approximation
factor and the diameter of the candidate trees. Specifically, we show three
results. First, we construct a plane tree $T_{\text{ALG}}$ with diameter at
most four that satisfies $|T_{\text{ALG}}|\ge \delta\cdot |T_{\text{OPT}}|$ for
$\delta>0.546$, thereby substantially improving the currently best known
approximation factor. Second, we show that the longest plane tree among those
with diameter at most three can be found in polynomial time. Third, for any
$d\ge 3$ we provide upper bounds on the approximation factor achieved by a
longest plane tree with diameter at most $d$ (compared to a longest general
plane tree).
