  A tanglegram $\cal T$ consists of two rooted binary trees with the same
number of leaves, and a perfect matching between the two leaf sets. In a
layout, the tanglegrams is drawn with the leaves on two parallel lines, the
trees on either side of the strip created by these lines are drawn as plane
trees, and the perfect matching is drawn in straight line segments inside the
strip. The tanglegram crossing number ${\rm cr}({\cal T})$ of $\cal T$ is the
smallest number of crossings of pairs of matching edges, over all possible
layouts of $\cal T$. The size of the tanglegram is the number of matching
edges, say $n$. An earlier paper showed that the maximum of the tanglegram
crossing number of size $n$ tanglegrams is $<\frac{1}{2}\binom{n}{2}$; but is
at least $\frac{1}{2}\binom{n}{2}-\frac{n^{3/2}-n}{2}$ for infinitely many $n$.
Now we make better bounds: the maximum crossing number of a size $n$ tanglegram
is at most $ \frac{1}{2}\binom{n}{2}-\frac{n}{4}$, but for infinitely many $n$,
at least $\frac{1}{2}\binom{n}{2}-\frac{n\log_2 n}{4}$. The problem shows
analogy with the Unbalancing Lights Problem of Gale and Berlekamp.
