  The circular descent of a permutation $\sigma$ is a set $\{\sigma(i)\mid
\sigma(i)>\sigma(i+1)\}$. In this paper, we focus on the enumerations of
permutations by the circular descent set. Let $cdes_n(S)$ be the number of
permutations of length $n$ which have the circular descent set $S$. We derive
the explicit formula for $cdes_n(S)$. We describe a class of generating binary
trees $T_k $ with weights. We find that the number of permutations in the set
$CDES_n(S)$ corresponds to the weights of $T_k$. As a application of the main
results in this paper, we also give the enumeration of permutation tableaux
according to their shape.
