  Forward-time models of diversification (i.e., speciation and extinction)
produce phylogenetic trees that grow "vertically" as time goes by. Pruning the
extinct lineages out of such trees leads to natural models for reconstructed
trees (i.e., phylogenies of extant species). Alternatively, reconstructed trees
can be modelled by coalescent point processes (CPP), where trees grow
"horizontally" by the sequential addition of vertical edges. Each new edge
starts at some random speciation time and ends at the present time; speciation
times are drawn from the same distribution independently. CPP lead to extremely
fast computation of tree likelihoods and simulation of reconstructed trees.
Their topology always follows the uniform distribution on ranked tree shapes
(URT). We characterize which forward-time models lead to URT reconstructed
trees and among these, which lead to CPP reconstructed trees. We show that for
any "asymmetric" diversification model in which speciation rates only depend on
time and extinction rates only depend on time and on a non-heritable trait
(e.g., age), the reconstructed tree is CPP, even if extant species are
incompletely sampled. If rates additionally depend on the number of species,
the reconstructed tree is (only) URT (but not CPP). We characterize the common
distribution of speciation times in the CPP description, and discuss incomplete
species sampling as well as three special model cases in detail: 1) extinction
rate does not depend on a trait; 2) rates do not depend on time; 3) mass
extinctions may happen additionally at certain points in the past.
