  The homogeneous reconstructed evolutionary process is a birth-death process
without observed extinct lineages. Each species evolves independently with the
same diversification rates (speciation rate $\lambda(t)$ and extinction rate
$\mu(t)$) that may change over time. The process is commonly applied to model
species diversification where the data are reconstructed phylogenies, e.g.,
trees reconstructed from present-day molecular data, and used to infer
diversification rates.
  In the present paper I develop the general probability density of a
reconstructed tree under any time-dependent birth-death process. I elaborate on
how to adapt this probability density if conditioned on survival of one or two
initial lineages, or having sampled $n$ species and show how to transform
between the probability density of a reconstructed and the probability density
of the speciation times.
  I demonstrate the use of the general time-dependent probability density
functions by deriving the probability density of a reconstructed tree under a
birth-death-shift model with explicit mass-extinction events. I enrich this
compendium by providing and discussing several special cases, including: the
pure birth process, the pure death process, the birth-death process and the
critical branching process. Thus, I provide here most of the commonly used
birth-death models in a unified framework (e.g., same condition and same data)
with common notation.
