  We study the largest values of the $r$th row of Stern's diatomic array. In
particular, we prove some conjectures of Lansing. Our main tool is the
connection between the Stern sequence, alternating binary expansions and
continuants. This allows us to reduce the problem of ordering the elements of
the Stern sequence to the problem of ordering continuants. We describe an
operation that increases the value of a continuant, allowing us to reduce the
problem of largest continuants to ordering continuants of very special shape.
Finally, we order these special continuants using some identities and
inequalities involving Fibonacci numbers.
