Ary tree rooted tree where each parent has most three children

(e) You are dividing 112 apples among 5 boxes. What is the smallest number of apples that could appear in the box with the most apples?

2. Let X be a set with n elements. Be sure to justify all your answers. (a) (3 points) How many reflexive relations are there on X?

and equivalence relations on that set.

4. (a) (5 points) Give a formula for the number of subgraphs of Kn that have exactly n vertices (and prove that it is correct)

8. A perfect binary tree of height h is a binary tree of height h with 2h terminal vertices.

(a) (5 points) Show that if T is a perfect binary tree of height h then the left and right subtrees of the root are each perfect binary trees of height h − 1.

(b) (5 points) Let tn be the number of nonisomorphic (as 3-ary trees) 3-ary trees with�n vertices, n−=0 1−i titjtn−1−(i+j)).
and by convention set t0 = 1.