#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

Question 1: Write the proof that a tree has n-1 edges 

Answer 1: The definition of a tree is that there are no cycles i.e. atleast one vertex only has one neighbour. Such a vertex is called leaf. 

Now we pull it out and we get a smaller tree with one less vertex and one less edge. We use induction for the smaller tree the number of vertices - number of edges = 1
The base case is that there is only one vertex has 0 edges
hence 1-0 = 1

Hence proved

Question 2: What is the OEIS A-number of this sequence?
Answer 2: A000272		Number of trees on n labeled nodes: n^(n-2) with a(0)=1.

Question 3
Is the sequence (number of labeled connected graphs with as many edges as vertices in the OEIS? What is the A number?

Answer 3: A057500		Number of connected labeled graphs with n edges and n nodes.

Question 4:
What is the smallest r such that ATreeSeq(30, r) is not in the OEIS?

Answer 4
r = 13


Question 5
What is teh ratio of the time it takes between time(TreeSeq(75)) and time (TreeSeq1(75))

Answer 5
65.74

Question 6
Let r_e(n) be the number of roots trees where every vertex has an even number of children

Answer 6
r_e(n)  =  x*(1+r_e(x)^2/2!+r_e(x)^4/4!+...
r_e(x) = x*cosh(r_e(x))
Question 7
How many labelled trees are there with 27 vertices and 6 leaves
147591129548061384922325901312000000
Question 8
Conjecture an explicit formula for the average number of leaves of a labeled tree with n vertices