We report the existence of a phenomenon of inheritance among periodic trajectories of the quadratic map. Within the set S_k of all possible trajectories of period k we find "mother” trajectories from which several "daughters” may be derived by simple polynomial transformations. For example, from the six orbital points z_i of a period-six mother we get three additional period-six daughters as the zeros of the six cubics x³ − 3x − z_i = 0. Daughters might have daughters. This stratification shows that periodic orbits are not necessarily independent of each other. This fact could be of importance for decomposing certain sums involving sets of periodic trajectories, particularly for trace formulas underlying semiclassical interpretations of spectra in atomic physics.