deletions | additions       diff --git a/subsubsection_Dendrograms_Dendrograms_are_hierarchical__.tex b/subsubsection_Dendrograms_Dendrograms_are_hierarchical__.tex  index 0a27524..9411601 100644  --- a/subsubsection_Dendrograms_Dendrograms_are_hierarchical__.tex  +++ b/subsubsection_Dendrograms_Dendrograms_are_hierarchical__.tex 
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%We refer the reader to Rosolowskly et al. (2008) for a complete description of applying Dendrograms to molecular cloud structures.  K15 K16  consider two dendrogram statistics: the number of featres features  or leaves and the histogram of leaf intensities. To compute the first statistic, we generate multiple dendrograms per output by varying $\delta_{\rm min}$ from $10^{-2.5}$ K$-$10$^{0.5}$ K in 100 logarithmic steps. We then count the total number of leaves associated with each $\delta_{\rm min}$. Figure 12 displays this statistic for the two fiducial outputs. Output W2T1t0.2 follows a pure power-law, while output T2t0 deviates from a power-law at $\delta_{\rm min}\sim$1 K. We find that the output with feedback contains more structure than the purely turbulent output at all scales.  {\bf RB(increasing past min delta creates a power law according to Burkhart et al. (2013), is this a counterexample?)}  The distribution of peak intensities provides additional insight into To compute  the dendrogram structure. We second statistic, we  create a series of dendrograms for the outputs using the same range of $\delta$, but instead of counting features we produce histograms of the leaf intensities for each value. %For each minimum delta, we create a Dendrogram, standardize its intensity leaves and branches, and generate histograms. We renormalize the intensities so that the mean of the histogram falls at zero.  Figure 12 displays the number of leaves as a function of $\delta$ for the two fiducial outputs. Output W2T1t0.2 follows a pure power-law, while output T2t0 deviates from a power-law at $\delta_{\rm min}\sim$1 K. The latter trend agrees with the results of \citet{burkhart13a}, who analyzed for MHD simulations of turbulence and found that the number of leaves significantly declines as $\delta$ increases. They also demonstrated that the power-law index for larger $\delta$ values steepens from -1.1 to -3.9 as sonic Mach number declines from 7 to 0.7. The appearance of a pure power-law for output W2T1t0.2 suggests an interesting signature of feedback; it increases the hierarchy significantly for large $\delta$. As a result, we also find that the output with feedback contains more structure than the purely turbulent output at all scales.  %{\bf RB(increasing past min delta creates a power law according to Burkhart et al. (2013), is this a counterexample?)} Yes!  The distribution of peak intensities provides additional insight into the emission structure.  Figure 13 shows superimposed histograms of the two fiducial outputs for all $\delta$. {\bf These histograms don't look very different for 100 steps in delta.} The two outputs produce yield  significantly different distributions. The histograms of T2t0 contain a wider range of leaf values than those of W1T2t0.2, whose histograms are all strongly peaked on the mean value. Output W1T2t0.2 also produces a long tail of high intensity values. % but many fewer negative ones. SSRO: note delta is not negative; log delta is negative Like the previous statistic, this indicates that feedback increases the amount of hierarchy in the emission.