  In this paper, we study the sign-changing radial solutions of the following
coupled Schr\"odinger system \begin{equation}
  \left\{
  \begin{array}{lr}
  -{\Delta}u_j+\lambda_j u_j=\mu_j u_j^3+\sum_{i\neq j}\beta_{ij} u_i^2 u_j
\,\,\,\,\,\,\,\, \mbox{in }B_1 ,\nonumber
  u_j\in H_{0,r}^1(B_1)\mbox{ for }j=1,\cdots,N.\nonumber
  \end{array}
  \right. \end{equation} Here, $\lambda_j,\,\mu_j>0$ and
$\beta_{ij}=\beta_{ji}$ are constants for $i,j=1,\cdots,N$ and $i\neq j$. $B_1$
denotes the unit ball in the Euclidean space $\mathbb{R}^3$ centred at the
origin. For any $P_1,\cdots,P_N\in\mathbb{N}$, we prove the uniqueness of the
radial solution $(u_1,\cdots,u_j)$ with $u_j$ changes its sign exactly $P_j$
times for any $j=1,\cdots,N$ in the following case: $\lambda_j\geq1$ and
$|\beta_{ij}|$ are small for $i,j=1,\cdots,N$ and $i\neq j$. New Liouville-type
theorems and boundedness results are established for this purpose.
