Five statistical questions about the tree of life

Stochastic modeling of phylogenies raises five questions that have received varying levels of attention from quantitatively inclined biologists. 1) How large do we expect (from the model) the ratio of maximum historical diversity to current diversity to be? 2) From a correct phylogeny of the extant species of a clade, what can we deduce about past speciation and extinction rates? 3) What proportion of extant species are in fact descendants of still-extant ancestral species, and how does this compare with predictions of models? 4) When one moves from trees on species to trees on sets of species (whether traditional higher order taxa or clades within PhyloCode), does one expect trees to become more unbalanced as a purely logical consequence of tree structure, without signifying any real biological phenomenon? 5) How do we expect that fluctuation rates for counts of higher order taxa should compare with fluctuation rates for number of species? We present a mathematician's view based on an oversimplified modeling framework in which all these questions can be studied coherently.