CS453 Assignment 6 Solved

Questions with a (⋆) are each worth 1 bonus point for 453 students.

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=∑𝑛 ; 2(𝑛−1)=totaldegree=∑𝑗𝑛. 𝑗𝑗

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

It suffices to show that 𝑛 𝑗

𝑛

∑(2−𝑗)𝑛 =2 𝑗

𝑗=1
= 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. Showthatif𝐿(𝑢)+𝐿(𝑣)=𝐷(𝑢,𝑣),then𝑟mustbeontheunique 𝑢, 𝑣-path.

c. Show that for any two vertices 𝑢 and 𝑣, 𝐷(𝑢, 𝑣) ≤ 2𝐻.
d. Showthatif𝐷(𝑢,𝑣)=2𝐻,then𝑢and𝑣mustbenon-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.
   1. Show that 𝑇 has 𝑏𝑞 edges for some integer 𝑏.
   2. Find a formula for the number of vertices of 𝑇 in terms of 𝑏, 𝑞.
   3. 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.
   1. Find tight lower and upper bounds for 𝐻, the height of 𝑇.
   2. 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.