#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:

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

#Attendance question 1
#Write the above proof if possible with more details in your own words
#Proof: A tree with n vertices must have n-1 edges
#A labeled tree is a connected graph without cycles because of that at least one vertex 
#must have only one neighbor. This is because if we have the below connected graph
#1-2-3
#then if 1 and 3 are connected to each other it would create a cycle so their must be at
#least one neighbor with only one neighbor. This is also called a leaf.
#If the lead is removed from the tree we still have a labeled tree and therefore bu induction the smaller tree has n-1 edges because n-edges is 1.
#Base case: a tree with one vertex with has n-1 edges (1-1= 0 edges) 

#Attendance Question 2
#What is the OEIS A-number of this sequence?
#Attendance Answer
#1,1,2,16,125,1296,16807
#A272

#Attendance Question 3
#Is this sequence (number of labelled connected graphs with as many edges as vertices
#in the OEIS
#Attendance Answer
#0,0,1,15,222,3660
#A57500

#Attendance Question 4
#What is the smallest r such that ATreeSeq(30,r) is not in the OEIS?
#Attendance Answer
#r=13

#Attendance Question 5
#what is the ratio of the time it takes between time(TreeSeg(75)); 
#and time(TreeSeq1(75))
#Attendance Answer
#84.44871795

#Attendance Question 6
#Let r_e(n) be the number of roots trees where every vertex has an even number of children 
#set up a functional equation for egf of r_e(n)
#Attendance Answer
#R_e(x)=Sum(r_e(n)*x^n/n!,n=0..infinity)
#   7
#  / \
# 5   9

#Attendance question 
#how many labelled trees are there with 27 vertices and 6 leaves
#Attendance Answer
#7776

#Attendance Question 
#Conjecture an explicit formula for the average number of leaves of a labelled tree with n vertices
#A bigger challenge: what is the limit if you divide by n
#An even bigger challenge: Prove the conjecture