k=3: family 9 (maximal 3: [['pm', 'pz', 'qm', 'qw', 'tm', 'sz'], ['pm', 'pz', 'qw', 'qz', 'tm', 'sz'], ['qm', 'qw', 'qz', 'tm', 'sz']]); subtrees 67; xhat in the hull of every subtree (exact): True
      H_k(xhat) = 8/3 = k-(k-2)/(2k-3): True;  LP: min over conv S = 3.000000, min over P and family hulls = 2.666667
k=4: family 9 (maximal 3: [['pm', 'pz', 'qm', 'qw', 'tm', 'sz'], ['pm', 'pz', 'qw', 'qz', 'tm', 'sz'], ['qm', 'qw', 'qz', 'tm', 'sz']]); subtrees 67; xhat in the hull of every subtree (exact): True
      H_k(xhat) = 18/5 = k-(k-2)/(2k-3): True;  LP: min over conv S = 4.000000, min over P and family hulls = 3.600000
k=5: family 9 (maximal 3: [['pm', 'pz', 'qm', 'qw', 'tm', 'sz'], ['pm', 'pz', 'qw', 'qz', 'tm', 'sz'], ['qm', 'qw', 'qz', 'tm', 'sz']]); subtrees 67; xhat in the hull of every subtree (exact): True
      H_k(xhat) = 32/7 = k-(k-2)/(2k-3): True;  LP: min over conv S = 5.000000, min over P and family hulls = 4.571429
k=6: family 9 (maximal 3: [['pm', 'pz', 'qm', 'qw', 'tm', 'sz'], ['pm', 'pz', 'qw', 'qz', 'tm', 'sz'], ['qm', 'qw', 'qz', 'tm', 'sz']]); subtrees 67; xhat in the hull of every subtree (exact): True
      H_k(xhat) = 50/9 = k-(k-2)/(2k-3): True;  LP: min over conv S = 6.000000, min over P and family hulls = 5.555556
k=7: family 9 (maximal 3: [['pm', 'pz', 'qm', 'qw', 'tm', 'sz'], ['pm', 'pz', 'qw', 'qz', 'tm', 'sz'], ['qm', 'qw', 'qz', 'tm', 'sz']]); subtrees 67; xhat in the hull of every subtree (exact): True
      H_k(xhat) = 72/11 = k-(k-2)/(2k-3): True;  LP: min over conv S = 7.000000, min over P and family hulls = 6.545455
k=8: family 9 (maximal 3: [['pm', 'pz', 'qm', 'qw', 'tm', 'sz'], ['pm', 'pz', 'qw', 'qz', 'tm', 'sz'], ['qm', 'qw', 'qz', 'tm', 'sz']]); subtrees 67; xhat in the hull of every subtree (exact): True
      H_k(xhat) = 98/13 = k-(k-2)/(2k-3): True;  LP: min over conv S = 8.000000, min over P and family hulls = 7.538462
