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In this paper, we study the stability of fractional Sobolev trace inequalitywithin both the functional and critical point settings.  In the functional setting, we establish the following sharp estimate:  $$C_{\mathrm{BE}}(n,m,\alpha)\inf_{v\in\mathcal{M}_{n,m,\alpha}}\left\Vertf-v\right\Vert_{D_\alpha(\mathbb{R}^n)}^2 \leq \left\Vertf\right\Vert_{D_\alpha(\mathbb{R}^n)}^2 - S(n,m,\alpha)\left\Vert\tau_mf\right\Vert_{L^{q}(\mathbb{R}^{n-m})}^2,$$  where $0\leq m< n$, $\frac{m}{2}<\alpha<\frac{n}{2},q=\frac{2(n-m)}{n-2\alpha}$ and $\mathcal{M}_{n,m,\alpha}$ denotes the manifoldof extremal functions. Additionally, We find an explicit bound for thestability constant $C_{\mathrm{BE}}$ and establish a compactness resultensuring the existence of minimizers.  In the critical point setting, we investigate the validity of a sharpquantitative profile decomposition related to the Escobar trace inequality andestablish a qualitative profile decomposition for the critical ellipticequation  \begin{equation*}  \Delta u= 0 \quad\text{in }\mathbb{R}_+^n,\quad\frac{\partial u}{\partialt}=-|u|^{\frac{2}{n-2}}u \quad\text{on }\partial\mathbb{R}_+^n.  \end{equation*}  We then derive the sharp stability estimate:  $$  C_{\mathrm{CP}}(n,\nu)d(u,\mathcal{M}_{\mathrm{E}}^{\nu})\leq \left\Vert\Delta u +|u|^{\frac{2}{n-2}}u\right\Vert_{H^{-1}(\mathbb{R}_+^n)},  $$  where $\nu=1,n\geq 3$ or $\nu\geq2,n=3$ and $\mathcal{M}_{\mathrm{E}}^\nu$represents the manifold consisting of $\nu$ weak-interacting Escobar bubbles.Through some refined estimates, we also give a strict upper bound for$C_{\mathrm{CP}}(n,1)$, which is $\frac{2}{n+2}$.

On the stability of fractional Sobolev trace inequality and corresponding profile decomposition