This paper concerns paleontological phylogenies that have a "budding" configuration, wherein "ancestral" species persist through branching events to coexist with their "descendants." Two principal tests are proposed for detecting patterns within such trees. The first test, called the "ancestor-descendant extinction test," compares the number of cases in which, after a split, the ancestral species became extinct before its descendant with the number of cases in which the descendant became extinct before its ancestor. The second test, called the "ancestor-descendant speciation test," compares the number of cases in which, after a split, the ancestral species gave rise to a further species with the number of cases in which the descendant species gave rise to a further species. The null hypothesis in each case is that the frequencies are equal, as predicted by a random Markovian branching model of evolution. Five stratophenetic species-level phylogenies of three taxonomic groups, planktonic foraminifera, nannofossils, and graptoloids, are examined using these tests, including one (Paleogene planktonic foraminifera) that is presented for the first time. In all cases, the phylogenetic trees are found to be strongly nonrandom. The general pattern, although by no means expressed perfectly in every case, corresponds to a Simpsonian "step-series," in which ancestor taxa are simultaneously more likely to become extinct and less likely to speciate than their coexisting descendants. It is shown that this pattern cannot simply be the result of simple age-dependent factors such as an increasing extinction risk in older taxa. Instead, the very fact that a species has given rise to another appears to increase its future extinction risk and decrease its likelihood of further speciation. Many possible biases may affect the shape of paleontological phylogenies, which are as yet poorly understood and unquantified. One potentially important effect follows from the taxonomic subdivision of gradual chronoclines into artificial morphospecies, such as might conceivably induce a step-series pattern in the phylogeny. Even if this is the partial or entire reason for the observed patterns, it would appear to imply directional evolution in phyletic gradualism. Other possible artifacts are discussed, but they are regarded as probably too weak to produce the observed patterns. If the pattern is not artificial, the fact that three of the best known fossil groups exhibit substantial asymmetries in speciation and extinction argues against the currently popular "nonprogressive" view of evolution. Instead, the evolutionary step-series pattern is consistent with the classical Darwinian concept of the general competitive superiority of newly evolved species over their ancestors and supports the idea of evolutionary progress.