#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

Q1. 
    Write the above prove with more details, in your own words [A tree with n verticies has n-1 edges]

A1. 
    Suppose a tree A. Based on definitions, A has no cycles. Thus at least there is one vertex to be a leaf node (which means it has only 1 neighbor). 
    If we remove this node, it still has no cycles, and removing this node does not break connectivity (since it only has one neighbor). So after removing this node the remaining is still a tree. At the end there will only be one vertex left, which has 0 edge. So we have removed n - 1 verticies and totally removed n - 1 edges. Thus a tree with n vertex has n - 1 edges. 

Q2. 
    What is the OEIS A number for this sequence? 

A2. 
    A272

Q3.
    is this sequence in OEIS? 

A3. 
    Yes, A57500

Q4. 
    What is the smallest r such that ATreeSeq(30, r) is not in OEIS? 

A4. 
    r = 13 is not in the OEIS 

Q5. 
    what is the ratio of the time it takes on your computer between time(TreeSeq1(75)) and time(TreeSeq(75))

A5. 
    60.973 vs 0.72 = 84.685:1

Q6.
    let r_e(n) be the number of roots trees where very vertex has an even number of children. set up a functional equation for egf of r_e(n) 
    R_e(x) = sum(r_e(n)*n^n/n!, n=0..inifnity)

    what are the first 20 terms of this sequence?  

A6. 
    Based on the methodology described in the class, we can keep removing root. 

    Which will give us R(x) = x*(1 + R(x)^2/2! + R(x)^4/4! + ... [only even numbers])

    And this should be equal to R(x) = x * cosh(R(x))

    L := FunEqToSeq(cosh(z), z, 20):
    seq(L[i]*factorial(i), i = 1 .. 20) = 1, 0, 3, 0, 65, 0, 3787, 0, 427905, 0, 79549811, 0, 22036379521, 0, 8513206310715, 0, 4374455745966593, 0, 2885264091484122979, 0. 

    This is not in the OEIS 

#### TODO? WHY X*? ####

Q7. 
    How many labelled trees are there with 27 vertices and 6 leaves 

A7. 
    L := TreeSeqL(27, t);
    coeff(L[27], t, 6) = 
                147591129548061384922325901312000000


Q8. 
    [CHALLENGE]
    Conjecture an explicit formula for the average number of leaves of a labelled tree with n verticies. 
c
    [Bigger CHALLENGE]
    what is the limit if you divide by the number of verticies. 

    [Even bigger CHALLENGE] 
    prove the conjecture. 
A8.