Suppose you are given as input a rooted tree t. a labeling of the vertices with labels from the set {1, 2, 3} is called valid if the label of every vertex is different from the label of its parent. the cost of a labeling is the number of vertices with smaller labels than their parents. design an efficient algorithm to find the minimum cost of a valid labeling of t.
prove that your algorithm is correct and analyze its running time.

Aprofessional hockey team has noticed that, over the past several seasons, as the number of wins increases, the total number of hot dogs sold at the concession stands increases. the hockey team concludes that winning makes people crave hot dogs. what is wrong with the hockey team's conclusion? a) it does not take into account the attendance during each season. b) it does not take into account the number of hamburgers sold during each season. c) it does not take into account the number of concession stands in the hockey arena. d) it does not take into account the number of seasons the hockey team has been in existence.