  Tanglegrams are formed by taking two rooted binary trees $T$ and $S$ with the
same number of leaves and uniquely matching each leaf in $T$ with a leaf in
$S$. They are usually represented using layouts, which embed the trees and the
matching of the leaves into the plane as in Figure 1. Given the numerous ways
to construct a layout, one problem of interest is the Tanglegram Layout
Problem, which is to efficiently find a layout that minimizes the number of
crossings. This parallels a similar problem involving drawings of graphs, where
a common approach is to insert edges into a planar subgraph. In this paper, we
will explore inserting edges into a planar tanglegram. Previous results on
planar tanglegrams include a Kuratowski Theorem, enumeration, and an algorithm
for drawing a planar layout. We start by building on these results and
characterizing all planar layouts of a planar tanglegram. We then apply this
characterization to construct a quadratic-time algorithm that inserts a single
edge optimally. Finally, we generalize some results to multiple edge insertion.
