  In this work, we completely characterize (i) permutation binomials of the
form $x^{{{2^n -1}\over {2^t-1}}+1}+ ax \in \mathbb{F}_{2^n}[x], n = 2^st, a
\in \mathbb{F}_{2^{2t}}^{*}$, and (ii) permutation trinomials of the form
$x^{2^s+1}+x^{2^{s-1}+1}+\alpha x \in \mathbb{F}_{2^t}[x]$, \end{enumerate}
where $s,t$ are positive integers. The first result, which was our primary
motivation, is a consequence of the second result. The second result may be of
independent interest.
