SII: additional mathematical material

Take \(n=8\) and hidden defect \(j=5\), whose binary digits are \(101\). One optimal tree uses the three queries \[A_0=\{1,3,5,7\},\qquad A_1=\{2,3,6,7\},\qquad A_2=\{4,5,6,7\}.\] The precise order may differ among equally optimal compiled trees. For this order, the supports are \[\{0,\ldots,7\}\xrightarrow{1}\{1,3,5,7\} \xrightarrow{0}\{1,5\}\xrightarrow{1}\{5\}.\] The terminal repair is \(r_5\). The path uses three diagnostic calls and one repair call. The complete tree stores all eight repair leaves and seven query nodes, not merely this path.