 Minimum vertex cover of a tree

Root a finite tree, and let $D_0,D_1$ be the optimum vertex cover sizes of a descendant subtree conditional on excluding or including its root. Excluding a parent forces all children to be included; including it permits either child state. These facts prove the displayed recurrences by structural induction. Empty child sums give $D_0=0,D_1=1$ at a leaf of a rooted tree. Evaluating in postorder traversal and taking the smaller root state gives a linear-time dynamic programming algorithm.