#ATTENDANCE QUIZ FOR LECTURE 22 of Dr. Z.'s Math454(02) Rutgers University

# Please Edit this .txt page with Answers

#Email  ShaloshBEkhad@gmail.com 
#Subject: p22
#with an attachment called
#p22FirstLast.txt
#(e.g. p22DoronZeilberger.txt)

#Right after finishing watching the lecture but no later than Nov. Dec. 1, 2020, 8:00pm


THE NUMBER OF ATTENDANCE QUESTIONS WERE: 6

PLEASE LIST ALL THE QUESTIONS FOLLOWED, AFTER EACH, BY THE ANSWER

1. Write the above proof (if possible with more details) in your own words
Lemma: A tree with n vertices must have n-1 edges
Proof: By definition, a tree is a connected graph without any cycles. In order for a graph to have no cycles, at least one vertex can only have one neighbor. We call this vertex a leaf. If this leaf is removed, we are left with a smaller tree with one less vertex and one less edge, where number of vertices-number of edges=1. For the base case, a tree with one vertex will have 0 edges, so 1-0=1. Therefore, a tree with n vertices has n-1 edges.

2. What is the OEIS A-number of this sequence?
1,1,3,16,125,1296,16807,262144,4782969
A000272

3. Is this sequence (number of labelled connected graphs with as many edges as vertices in the OEIS? What is the A-number?
0,0,1,15,222,3660,68295,1436568,33779340,880107840
A057500

4. What is the smallest r such that ATreeSeq(30,r) is not in the OEIS?
r=13

5. What is the ratio of the time it takes between time(TreeSeq(75)); and time(TreeSeq1(75));
47.250 : 0.562

6. Let r_e(n) be the number of roots trees where every vertex has an even number of childen. 
Set up a function equation for egf of r_e(n)
R_e(x)=Sum(r_e(n)*x^n/n!,n=0..infinity)
What are the first 20 terms of this sequence?
Is it in the OEIS?
1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420
A000108