  W. Mader [J. Graph Theory 65 (2010), 61--69] conjectured that for any tree
$T$ of order $m$, every $k$-connected graph $G$ with
$\delta(G)\geq\lfloor\frac{3k}{2}\rfloor+m-1$ contains a tree $T'\cong T$ such
that $G-V(T')$ remains $k$-connected. In 2010, Mader confirmed the conjecture
for the $k$-connected graph if $T$ is a path; very recently, Liu et al.
confirmed the conjecture if $k=2,3$. The conjecture is open for $k\geq 4$ till
now. In this paper, we show that Mader's conjecture is true for the
$k+1$-connected graph if $T$ is a spider and $\Delta(G)=|G|-1$.
