## 7 Analysis
### The Size of the Smaller Component  
In the previous section, we saw that HDT(0) scanned relatively few non-tree edges when used in either FCT(0) or FPP(0). Tables 12-16 show one reason for this: the smaller component was often _very_ small. In fact, the small component often contained just one or two nodes. The tables show the number of times the smaller subtree resulting from an edge deletion had at most two nodes (\"Num. \(\leq 2\)\") and more than two nodes (\"Num. \(>2\)\") for different types of input profiles. In all cases, subtrees of a size at most two outnumbered the rest by a wide margin. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline  & **5\%** & **10\%** & **15\%** & **20\%** & **25\%** & **30\%** & **35\%** & **40\%** & **45\%** & **50\%** \\ \hline Low & 69,133 & 65,415 & 61,975 & 57,305 & 53,638 & 47,722 & 44,590 & 40,668 & 35,545 & 31,668 \\ Medium & 461,875 & 420,106 & 372,561 & 400,387 & 387,020 & 353,904 & 334,324 & 292,983 & 250,507 & 251,315 \\ High & 42,210 & 32,766 & 41,489 & 36,797 & 40,709 & 29,770 & 29,068 & 29,206 & 26,317 & 26,811 \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Number of non-tree edges scanned by FPP(0) for matrices of order of 300 \(\times\) 6000 with different density levels. Figure 11: Number of non-tree edge FCT scans for profiles of binary phylogenetic trees. The blue curve corresponds to varying \(k\), while keeping \(n\) fixed at 100. The orange curve corresponds to varying \(n\) while keeping \(k\) fixed at 100. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline  & **5\%** & **10\%** & **15\%** & **20\%** & **25\%** & **30\%** & **35\%** & **40\%** & **45\%** & **50\%** \\ \hline Num. \(\leq 2\) & 31,130 & 30,709 & 31,054 & 30,443 & 30,194 & 29,359 & 29,393 & 28,839 & 27,603 & 26,696 \\ Num. \(>2\) & 882 & 908 & 905 & 907 & 901 & 908 & 892 & 912 & 900 & 879 \\ \hline \hline \end{tabular}
\end{table} 
Table 16: FPP(0): Number of subtrees of a size at most two versus number of subtrees of size greater than two for high-density matrices of order \(300\times 6000\). \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \(M_{\mathcal{P}}\) & **10,000** & **20,000** & **30,000** & **40,000** & **50,000** & **60,000** & **70,000** & **80,000** & **90,000** & **100,000** \\ \hline Num. \(\leq 2\) & 7453 & 14,985 & 22,566 & 29,957 & 37,532 & 44,959 & 52,500 & 59,630 & 66,984 & 74,780 \\ Num. \(>2\) & 667 & 1156 & 1648 & 2104 & 2555 & 3201 & 3460 & 3966 & 4643 & 4975 \\ \hline \hline \end{tabular}
\end{table} 
Table 13: FCT(0): Number of subtrees of a size at most two versus number of subtrees of size greater than two for profiles of binary trees with varying number of trees and 100 taxa. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \(M_{\mathcal{P}}\) & **10,000** & **20,000** & **30,000** & **40,000** & **50,000** & **60,000** & **70,000** & **80,000** & **90,000** & **100,000** \\ \hline
1 & 7012 & 14,188 & 21,324 & 28,550 & 35,603 & 42,883 & 49,848 & 56,924 & 64,181 & 71,334 \\
2 & 461 & 910 & 1387 & 1853 & 2300 & 2753 & 3235 & 3714 & 4155 & 4612 \\
3 & 158 & 287 & 434 & 586 & 721 & 862 & 1016 & 1154 & 1288 & 1448 \\
4 & 88 & 157 & 239 & 315 & 393 & 472 & 560 & 615 & 689 & 794 \\
5 & 56 & 101 & 149 & 197 & 242 & 290 & 350 & 392 & 431 & 495 \\
6 & 38 & 70 & 102 & 137 & 166 & 197 & 238 & 266 & 294 & 348 \\
7 & 27 & 48 & 71 & 101 & 125 & 143 & 174 & 190 & 218 & 245 \\
8 & 20 & 37 & 54 & 78 & 95 & 113 & 132 & 151 & 167 & 182 \\ \(>\)8 & 296 & 640 & 988 & 1308 & 1645 & 1970 & 2289 & 2626 & 2930 & 3231 \\ \hline \hline \end{tabular}
\end{table} 
Table 14: FCT(0): Number of subtrees of a size at most two versus number of subtrees of size greater than two for profiles of binary trees with varying number of taxa and 100 trees.