#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


1. Write the above proof in your own words. 


Given a labeled tree is a connected graph without cycles, it must be the case that there is at least one vertex with only one neighbor. We call this vertex a leaf. 


If this leaf is removed from the tree, a vertex is removed thereby removing an edge from the connected graph. 


In both cases, for the large tree and the small tree, the number of vertices - number of edges = 1.  


This also holds true for a graph with one vertice, as 1 - 0 = 1. 


Therefore, a tree with n vertices must have n-1 edges. 


2. What is the OEIS number of this sequence?


A000272


3. Is this sequence in the OEIS? What is the A number? 


Yes - A057500


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


R = 4


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


8427/91


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


r_e(x) = sum(r_e(n)*x^n/n!,n=0..infinity) 


What are the first 20 terms? Is it in the OEIS? 


r_e(x) = r_e(0) + r_e(1)*x + r_e(2)*x^2/2 + r_e(3)*x^3/6 + r_e(4)*x^4/24 + r_e(5)*x^5/120 + r_e(6)*x^6/720 + r_e(7)*x^7/5040 + r_e(8)*x^8/40320 + r_e(9)*x^9/362880 + r_e(10)*x^10/3628800 + r_e(11)*x^11/39916800 + r_e(12)*x^12/479001600 + r_e(13)*x^13/6227020800 + r_e(14)*x^14/87178291200 + r_e(15)*x^15/1307674368000 + r_e(16)*x^16/20922789888000 + r_e(17)*x^17/355687428096000 + r_e(18)*x^18/6402373705728000 + r_e(19)*x^19/121645100408832000 + r_e(20)*x^20/2432902008176640000


7. How many labeled trees are there with 27 vertices and 6 leaves?


79044165842816694716044003499779078685850599424


8. Conjecture an explicit formula for the average number of leaves of a labeled tree with n vertices. 


n^n-2