  We determine the paint cost spectrum for perfect $k$-ary trees.
  A coloring of the vertices of a graph $G$ with $d$ colors is said to be
\emph{$d$-distinguishing} if only the trivial automorphism preserves the color
classes. The smallest such $d$ is the distinguishing number of $G$ and is
denoted $\mbox{dist}(G).$ The \emph{paint cost of $d$-distinguishing $G$},
denoted $\rho^d(G)$, is the minimum size of the complement of a color class
over all $d$-distinguishing colorings. A subset $S$ of the vertices of $G$ is
said to be a \emph{fixing set} for $G$ if the only automorphsim that fixes the
vertices in $S$ pointwise is the trivial automorphism. The cardinality of a
smallest fixing set is denoted $\mbox{fix}(G)$. In this paper, we explore the
breaking of symmetry in perfect $k$-ary trees by investigating what we define
as the \emph{paint cost spectrum} of a graph $G$: $(\mbox{dist}(G);
\rho^{\mbox{dist}(G)}(G), \rho^{\mbox{dist}(G)+1}(G), \dots,
\rho^{\mbox{fix}(G)+1}(G))$ and the \emph{paint cost ratio} of $G$, which is
defined to be the fraction of paint costs in the paint cost spectrum equal to
$\mbox{fix}(G)$. We determine both the paint cost spectrum and the paint cost
ratio completely for perfect $k$-ary trees.
  We also prove a lemma that is of interest in its own right: given an
$n$-tuple, $n \geq 2$ of distinct elements of an ordered abelian group and $1
\leq k \leq n! -1$, there exists a $k \times n$ row permuted matrix with
distinct column sums.
