1. Show that for any 𝑘 ≥ 3, if a tree 𝑇 has fewer than 𝑘 leaves, then the maximum degree Δ(𝑇) among the vertices of 𝑇 must satisfy Δ(𝑇) < 𝑘. It can help to consider the summations
𝑛 𝑛𝑗 ; 2(𝑛 − 1) = total degree .
𝑗=1 𝑗=1
The phrase “𝑇 has fewer than 𝑘 leaves” means 𝑛1 < 𝑘.
The two sums can be combined into the single sum
𝑛
∑(2 − 𝑗)𝑛𝑗 = 2
𝑗=1
It suffices to show that 𝑛𝑗 = 0 for all 𝑗 ≥ 𝑘.
2. Let (𝑇, 𝑟) be a rooted tree. Recall that the level of a vertex 𝑥 is 𝐿(𝑥) = 𝐷(𝑟, 𝑥). Also, the height of a rooted tree 𝐻 is the maximum of the levels of its vertices.
a. Show that if 𝑟 is on the unique 𝑢, 𝑣-path, then 𝐷(𝑢, 𝑣) = 𝐿(𝑢) +
𝐿(𝑣).
b. Show that if 𝐿(𝑢) + 𝐿(𝑣) = 𝐷(𝑢, 𝑣), then 𝑟 must be on the unique 𝑢, 𝑣-path.
c. Show that for any two vertices 𝑢 and 𝑣, 𝐷(𝑢, 𝑣) ≤ 2𝐻.
d. Show that if 𝐷(𝑢, 𝑣) = 2𝐻, then 𝑢 and 𝑣 must be non-parents. Equivalently, you can show that if either 𝑢 or 𝑣 is a parent, then 𝐷(𝑢, 𝑣) < 2𝐻.
3. Suppose (𝑇, 𝑟) is a rooted 𝑞-ary tree where every parent has exactly 𝑞 children; such a tree is said to be saturated.
a. Show that 𝑇 has 𝑏𝑞 edges for some integer 𝑏.
b. Find a formula for the number of vertices of 𝑇 in terms of 𝑏, 𝑞.
c. Find a formula for the number of non-parents in terms of 𝑏, 𝑞.
4. Suppose (𝑇, 𝑟) is a rooted tree with exactly 1012 edges. Recall that a lower bound or an upper bound on 𝐻 is tight if there exists an example 𝑇 where that bound is attained.
a. Find tight lower and upper bounds for 𝐻, the height of 𝑇.
b. Find tight lower and upper bounds for 𝐻 if 𝑇 is a saturated rooted binary tree. Recall that saturated means every parent has the maximum allowed number of children; here, that number is 2.
