  Mader conjectured that for any tree $T$ of order $m$, every $k$-connected
graph $G$ with minimum degree at least $\lfloor\frac{3k}{2}\rfloor +m-1$
contains a subtree $T'\cong T$ such that $G-V(T')$ is $k$-connected. In this
paper, we give a characterization for a subgraph to contain an embedding of a
specified tree avoiding some vertex. As a corollary, we confirm Mader's
conjecture for $k\leq3$.
