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The homogeneous reconstructed evolutionary process is a birth-death processwithout observed extinct lineages. Each species evolves independently with thesame diversification rates (speciation rate $\lambda(t)$ and extinction rate$\mu(t)$) that may change over time. The process is commonly applied to modelspecies diversification where the data are reconstructed phylogenies, e.g.,trees reconstructed from present-day molecular data, and used to inferdiversification rates.  In the present paper I develop the general probability density of areconstructed tree under any time-dependent birth-death process. I elaborate onhow to adapt this probability density if conditioned on survival of one or twoinitial lineages, or having sampled $n$ species and show how to transformbetween the probability density of a reconstructed and the probability densityof the speciation times.  I demonstrate the use of the general time-dependent probability densityfunctions by deriving the probability density of a reconstructed tree under abirth-death-shift model with explicit mass-extinction events. I enrich thiscompendium by providing and discussing several special cases, including: thepure birth process, the pure death process, the birth-death process and thecritical branching process. Thus, I provide here most of the commonly usedbirth-death models in a unified framework (e.g., same condition and same data)with common notation.

The time-dependent reconstructed evolutionary process with a key-role for mass-extinction events