  We show that for each $k\in \mathbb{N}$, a link $L\subset S^3$ bounds a
degree $k$ Whitney tower in the 4-ball if and only if it is
\emph{$C_k$-concordant} to the unlink. This means that $L$ is obtained from the
unlink by a finite sequence of concordances and degree $k$ clasper surgeries.
In our construction the trees associated to the Whitney towers coincide with
the trees associated to the claspers.
  As a corollary to our previous obstruction theory for Whitney towers in the
4-ball, it follows that the $C_k$-concordance filtration of links is classified
in terms of Milnor invariants, higher-order Sato-Levine and Arf invariants.
  Using a new notion of $k$-repeating twisted Whitney towers, we also classify
a natural generalization of the notion of link homotopy, called twisted
\emph{self $C_k$-concordance}, in terms of $k$-repeating Milnor invariants and
$k$-repeating Arf invariants.
