  Let $\Omega\subseteq \mathbb{R}^{4}$ be a bounded domain with smooth boundary
$\partial\Omega$. In this paper, we establish the following sharp form of the
trace Adams' inequality in $W^{2,2}(\Omega)$ with
  zero mean value and zero Neumann boundary condition: \begin{equation*}
S({\alpha})=\underset{\int_{\Omega}udx=0,\frac{\partial
u}{\partial\nu}|_{\partial\Omega}=0,\Vert\Delta u\Vert_{2}\leq{1}}{\underset
{u\in{W^{2,2}(\Omega)\setminus\{0\}}}{\sup}}\int_{\partial \Omega} e^{\alpha
u^{2}}d\sigma<\infty \end{equation*} holds if and only if $ \alpha\leq12\pi^2$.
  Moreover, we prove a classification theorem for the solutions of a class of
nonlinear boundary value problem of bi-harmonic equations on the half space
$\mathbb{R}^4_{+}$. With this classification result, we can show that
$S({12\pi^2})$ is attained by using the blow-up analysis and capacitary
estimate. As an application, we prove a sharp trace Adams-Onofri type
inequality in general four dimensional bounded domains with smooth boundary.
