$\mathcal{Q}^{+}$: Characterising the structure of young star clusters
  Many young star clusters appear to be fractal, i.e. they appear to be
concentrated in a nested hierarchy of clusters within clusters. We present a
new algorithm for statistically analysing the distribution of stars to quantify
the level of sub-structure. We suggest that, even at the simplest level, the
internal structure of a fractal cluster requires the specification of three
parameters. (i) The 3D fractal dimension, $\mathcal{D}$, measures the extent to
which the clusters on one level of the nested hierarchy fill the volume of
their parent cluster. (ii) The number of levels, $\mathcal{L}$, reflects the
finite ratio between the linear size of the large root-cluster at the top of
the hierarchy, and the smallest leaf-clusters at the bottom of the hierarchy.
(iii) The volume-density scaling exponent, $\mathcal{C}=-\textrm{d}\ln[\delta
n]/\textrm{d}\ln[L]$ measures the factor by which the excess density, $\delta
n$, in a structure of scale $L$, exceeds that of the background formed by
larger structures; it is similar, but not exactly equivalent, to the exponent
in Larson's scaling relation between density and size for molecular clouds. We
describe an algorithm which can be used to constrain the values of $({\cal
D},{\cal L},{\cal C})$ and apply this method to artificial and observed
clusters. We show that this algorithm is able to reliably describe the three
dimensional structure of an artificial star cluster from the two dimensional
projection, and quantify the varied structures observed in real and simulated
clusters.
