Star-formation within galaxies appears on multiple scales , from spiral structure , to OB associations , to individual star clusters , and often sub-structure within these clusters .
This multitude of scales calls for objective methods to find and classify star-forming regions , regardless of spatial size .
To this end , we present an analysis of star-forming groups in the local group spiral galaxy M33 , based on a new implementation of the Minimum Spanning Tree ( MST ) method .
Unlike previous studies which limited themselves to a single spatial scale , we study star-forming structures from the effective resolution limit ( \sim 20 pc ) to kpc scales .
Once the groups are identified , we study their properties , e.g .
size and luminosity distributions , and compare them with studies of young star clusters and giant molecular clouds ( GMCs ) .
We find evidence for a continuum of star-forming group sizes , which extends into the star-cluster spatial scale regime .
We do not find a characteristic scale for OB associations , unlike that found in previous studies , and we suggest that the appearance of such a scale was caused by spatial resolution and selection effects .
The luminosity function of the groups is found to be well represented by a power-law with an index , -2 , the same as has been found for the luminosity and mass functions of young star clusters , as well as the mass function of GMCs .
Additionally , the groups follow a similar mass-radius relation as GMCs .
The size distribution of the groups is best described by a log-normal distribution , the peak of which is controlled by the spatial scale probed and the minimum number of sources used to define a group .
We show that within a hierarchical distribution , if a scale is selected to find structure , the resulting size distribution will have a log-normal distribution .
We find an abrupt drop of the number of groups outside a galactic radius of \sim 4 kpc ( although individual high-mass stars are found beyond this limit ) , suggesting a change in the structure of the star-forming ISM , possibly reflected in the lack of GMCs beyond this radius .
Finally , we find that the spatial distribution of H ii regions , GMCs , and star-forming groups are all highly correlated .