  We prove that $\det A\leq 6^\frac{n}{6}$ whenever $A\in\{0,1\}^{n\times n}$
contains at most $2n$ ones. We also prove an upper bound on the determinant of
matrices with the $k$-consecutive ones property, a generalisation of the
consecutive ones property, where each row is allowed to have up to $k$ blocks
of ones. Finally, we prove an upper bound on the determinant of a path-edge
incidence matrix in a tree and use that to bound the leaf rank of a graph in
terms of its order.
