Answer the following questions about the rooted tree shown below.  For questions in which you need to list more than one vertex, list the vertices in alphabetical order and separate them with a comma (no spaces).  1. What are the ancestor(s) of Q? 2. List the leaves of the graph.  3. How many internal vertices does graph have?

Use a tree to model the parent-child relationships in the following family. Paloma has three children: Sherry, Laura, and Kerry. Sherry has one child, Donna. Laura has one child, Betty. Kerry has one child, Hopalong. Use vertices to model the people and edges to represent the parent-child relationships. Label the vertices with the person's initial. Choose the graph that represents the parent-child relationships. O A. O C. K L 14 L K Au P H B D S M B. P HBD S Au H B D S O D. K L P • C H B D S

Recursively, a complete binary tree is a graph. Basis step: One vertex forms a binary tree. Inductive step: If T1 and T2 are disjoint complete binary trees with roots r1, r2, respectively, the graph produced by starting with a root r and adding an edge from r to each of the vertices r1, r2 is also complete. Recursively define a complete binary tree's leaves. Base step: The full binary tree has one root, r. Inductive step: T's leaves are the union of T1's and T2's.       Class defines binary tree height h(T). Use structural induction to prove that `(T), the number of leaves of a complete binary tree T, is less than 2 h(T).

a. Draw a graph with 5 vertices and 5 edges. Can such a graph be a tree? Carefully explain your answer.