The stellar initial mass functions (IMFs) for the Galactic bulge, the Milky Way, other galaxies, clusters of galaxies, and the integrated stars in the universe are composites from countless individual IMFs in star clusters and associations where stars form. These galaxy-scale IMFs, reviewed in detail here, are not steeper than the cluster IMFs except in rare cases. This is true even though low-mass clusters generally outnumber high-mass clusters and the average maximum stellar mass in a cluster scales with the cluster mass. The implication is that the mass distribution function for clusters and associations is a power law with a slope of -2 or shallower. Steeper slopes, even by a few tenths, upset the observed equality between large- and small-scale IMFs. Such a cluster function is expected from the hierarchical nature of star formation, which also provides independent evidence for the IMF equality when it is applied on subcluster scales. We explain these results with analytical expressions and Monte Carlo simulations. Star clusters appear to be the relaxed inner parts of a widespread hierarchy of star formation and cloud structure. They are defined by their own dynamics rather than by preexisting cloud boundaries.
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