  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 sub-cluster 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 pre-existing cloud
boundaries.
