  Let $q$ be a power of $2$. Recently, Tu and Zeng considered trinomials of the
form $f(X)=X+aX^{(1/4)q^2(q-1)}+bX^{(3/4)q^2(q-1)}$, where $a,b\in\Bbb
F_{q^2}^*$. They proved that $f$ is a permutation polynomial of $\Bbb F_{q^2}$
if $b=a^{2-q}$ and $X^3+X+a^{-1-q}$ has no root in $\Bbb F_q$. In this paper,
we show that the above sufficient condition is also necessary.
