In this paper, we study factorizations of cycles. The main result is that under
certain condition, the number of ways to factor a $d$-cycle into a product of cycles of
prescribed lengths is $d^{r-2}.$ To prove our result, we first define a new class of
combinatorial objects, multi-noded rooted trees, which generalize rooted trees. We find the
cardinality of this new class which with proper parameters is exactly $d^{r-2}.$ The main
part of this paper is the proof that there is a bijection from factorizations of a
$d$-cycle to multi-noded rooted trees via factorization graphs. This implies the desired
formula. The factorization problem we consider has its origin in geometry, and is related
to the study of a special family of Hurwitz numbers: pure-cycle Hurwitz numbers. Via the
standard translation of Hurwitz numbers into group theory, our main result is equivalent to
the following: when the genus is $0$ and one of the ramification indices is $d,$ the degree
of the covers, the pure-cycle Hurwitz number is $d^{r-3},$ where $r$ is the number of
branch points.