Forward-in-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. CPPs 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-in-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.