This problem asks about binary tree preorder traversal. In preorder traversal, we visit root first, then left subtree, then right subtree. If nodes P and Q appear consecutively as PQ, we need to determine which parent-child relationships are impossible.
Let's analyze case five: Q is P's father. In preorder traversal, we always visit the parent before its children. If Q is P's father, then Q must be visited before P, not after P. Therefore, having P followed immediately by Q is impossible when Q is P's father.
Now let's examine case one: P is Q's cousin. Cousins have different parents who are siblings. In preorder traversal, we visit the entire left subtree before moving to the right subtree. For cousins P and Q to be consecutive, the uncle node must come between them, making PQ impossible.
Let's verify the possible cases. Case two: P and Q are siblings with P as left child and Q as right child - this gives us Parent, P, Q order. Case four: P is Q's father - this gives us P, Q order when Q is the left child. Case three: Q is P's father's sibling - after visiting P's subtree, we visit uncle Q next.
In conclusion, we have analyzed all five cases. Cases one and five are impossible: cousins cannot be consecutive because the uncle must come between them, and a child cannot appear before its parent in preorder traversal. Cases two, three, and four are all possible. Therefore, the answer is C: cases one and five are impossible.