  The spatial properties of small star-clusters suggest that they may originate
from a fragmentation cascade of the cloud for which there might be traces up to
a few dozen of kAU. Our goal is to investigate the multi-scale spatial
structure of gas clumps, to probe the existence of a hierarchical cascade and
to evaluate its possible link with star production in terms of multiplicity.
From the Herschel emission maps of NGC 2264, clumps are extracted using getsf
software at each of their associated spatial resolution, respectively [8.4,
13.5, 18.2, 24.9, 36.3]". Using the spatial distribution of these clumps and
the class 0/I Young Stellar Object (YSO) from Spitzer data, we develop a
graph-theoretic analysis to represent the multi-scale structure of the cloud as
a connected network. From this network, we derive three classes of multi-scale
structure in NGC 2264 depending on the number of nodes produced at the deepest
level: hierarchical, linear and isolated. The structure class is strongly
correlated with the column density $N_{\rm H_2}$ since the hierarchical ones
dominate the regions whose N$_{\rm H_2} > 6 \times 10^{22}$cm$^{-2}$. Although
the latter are in minority, they contain half of the class 0/I YSOs proving
that they are highly efficient in producing stars. We define a novel
statistical metric, the fractality coefficient F that measure the fractal index
describing the scale-free process of the cascade. For NGC 2264, we estimate F =
1.45$\pm$0.12. However, a single fractal index fails to fully describe a
scale-free process since the hierarchical cascade starts at a 13 kAU
characteristic spatial scale. Our novel methodology allows us to correlate YSOs
with their multi-scale gaseous environment. This hierarchical cascade that
drives efficient star formation is suspected to be both hierarchical and rooted
by the larger-scale gas environment up to 13 kAU.
