  We introduce a natural way to construct a random subset of a homogeneous
Cantor set $C$ in $[0,1]$ via random labelings of an infinite $M$-ary tree,
where $M\geq 2$. The Cantor set $C$ is the attractor of an equicontractive
iterated function system $\{f_1,\dots,f_N\}$ that satisfies the open set
condition with $(0,1)$ as the open set. For a fixed probability vector
$(p_1,\dots,p_N)$, each edge in the infinite $M$-ary tree is independently
labeled $i$ with probability $p_i$, for $i=1,2,\dots,N$. Thus, each infinite
path in the tree receives a random label sequence of numbers from
$\{1,2,\dots,N\}$. We define $F$ to be the (random) set of those points $x\in
C$ which have a coding that is equal to the label sequence of some infinite
path starting at the root of the tree.
  The set $F$ may be viewed as a statistically self-similar set with extreme
overlaps, and as such, its Hausdorff and box-counting dimensions coincide. We
prove non-trivial upper and lower bounds for this dimension, and obtain the
exact dimension in a few special cases. For instance, when $M=N$ and $p_i=1/N$
for each $i$, we show that $F$ is almost surely of full Hausdorff dimension in
$C$ but of zero Hausdorff measure in its dimension.
  For the case of two maps and a binary tree, we also consider deterministic
labelings of the tree where, for a fixed integer $m\geq 2$, every $m$th edge is
labeled $1$, and compute the exact Hausdorff dimension of the resulting subset
of $C$.
