  The site frequency spectrum (SFS) is a popular summary statistic of genomic
data. While the SFS of a constant-sized population undergoing neutral mutations
has been extensively studied in population genetics, the rapidly growing amount
of cancer genomic data has attracted interest in the spectrum of an
exponentially growing population. Recent theoretical results have generally
dealt with special or limiting cases, such as considering only cells with an
infinite line of descent, assuming deterministic tumor growth, or sending the
tumor size to infinity. In this work, we we derive exact expressions for the
expected SFS of a cell population that evolves according to a stochastic
branching process, first for cells with an infinite line of descent and then
for the total population, evaluated either at a fixed time (fixed-time
spectrum) or at the stochastic time at which the population reaches a certain
size (fixed-size spectrum). We find that while the rate of mutation scales the
SFS of the total population linearly, the rates of cell birth and cell death
change the shape of the spectrum at the small-frequency end, inducing a
transition between a $1/j^2$ power-law spectrum and a $1/j$ spectrum as cell
viability decreases. We use this insight to propose a simple estimator for the
ratio between the rate of cell death and cell birth, that we show to accurately
recover the true ratio from synthetic single-cell sequencing data. Although the
discussion is framed in terms of tumor dynamics, our results apply to any
exponentially growing population of individuals undergoing neutral mutations.
