Show that the single-linkage method is equivalent to taking \(\alpha_{i}=\alpha_{j}=0.5, \beta=0\), and \(\gamma=-0.5\) in the Lance-Williams formula; the complete-linkage method is equivalent to \(\alpha_{i}=\alpha_{j}=0.5\), \(\beta=0\), and \(\gamma=0.5\); and the Ward's criterion is equivalent to \(\alpha_{i}=\frac{n_{i}+n_{k}}{n_{i}+n_{j}+n_{k}}, \alpha_{j}=\frac{n_{j}+n_{k}}{n_{i}+n_{j}+n_{k}}\), \(\beta=-\frac{n_{k}}{n_{i}+n_{j}+n_{k}}\), and \(\gamma=0\).