> #Attendence Q1:
> Write the prove of lemma 1 in your own words with more details.
> #Since the tree has no cycles, there is at least one vertex that has only one neighbor. Such a vertex is called a leaf. 
> #If you remove it, you get a smaller tree with one less vertex and one less edge, 
> #By the induction, for the smaller tree, the number of vertices - number of edges = 1
> #Base case, the tree with one vertex has 0 edges, 1-0=1
> #Hence proved the lemma: A tree with n vertices must have n-1 edges 
> 
> 
> #Attendence Q2:
> #What is the OEIS A number of this sequence? 
> read `M22.txt`;
> L:=TreeSeq(10);
L := [1, 1, 3, 16, 125, 1296, 16807, 262144, 4782969, 100000000]
> #A000272: number of trees of n labeled nodes.
> 
> #Attendence Q3:
> #Is this sequence in the OEIS? A number? (number of labeled connected graphs with as many edges as vertices)
> ATreeSeq(10,1);
[0, 0, 1, 15, 222, 3660, 68295, 1436568, 33779340, 880107840]
> #A057500: number of connected labeled graphs with n edges and n graphs
> 
> #Attendence Q4:
> #What is the smallest r such that ATreeSeq(30,r) is not in the OEIS?
> ATreeSeq(10,4);
[0, 0, 0, 0, 45, 4945, 331506, 18602136, 974679363, 50088981600]
> #Optional: submit it.
> ATreeSeq(10,5);
[0, 0, 0, 0, 10, 2997, 343140, 28044072, 1969994376, 128916045720]
> ATreeSeq(10,6);
[0, 0, 0, 0, 1, 1365, 290745, 35804384, 3431889000, 288982989000]
> ATreeSeq(10,7);
[0, 0, 0, 0, 0, 455, 202755, 39183840, 5228627544, 573177986865]
> ATreeSeq(10,8);
[0, 0, 0, 0, 0, 105, 116175, 37007656, 7032842901, 1016662746825]
> ATreeSeq(10,9);
[0, 0, 0, 0, 0, 15, 54257, 30258935, 8403710364, 1624745199910]
> ATreeSeq(10,10);
[0, 0, 0, 0, 0, 1, 20349, 21426300, 8956859646, 2352103292070]
> ATreeSeq(10,11);
[0, 0, 0, 0, 0, 0, 5985, 13112470, 8535294180, 3096620034795]
> ATreeSeq(10,12);
[0, 0, 0, 0, 0, 0, 1330, 6905220, 7279892361, 3717889913655]
> ATreeSeq(10,13);
[0, 0, 0, 0, 0, 0, 210, 3107937, 5557245480, 4078716030900]
> #n vertices with n+12 vertices is not in the OEIS. submitted.
> 
> #Attendence Q5:
> #What is the ratio of the time(TreeSeq(75)) and time(TreeSeq1(75))
> time(TreeSeq(75));
49.890
> time(TreeSeq1(75));
.703
> 49.890/0.703;
70.96728307
> 
> #Attendence Q6:
> #Let r_e(n) be the number of roots trees where every vertex has an even number of children. Set up an generating function for r_e(n).
> 
> f:= sum(r_e(n)*x^n/n!, n=0..infinity);
f := sum(r_e(n)*x^n/factorial(n), n = 0 .. infinity)
> F:=LIF(exp(z),z,10);
F := [1, 1, 3/2, 8/3, 125/24, 54/5, 16807/720, 16384/315, 531441/4480, 156250/567]
> seq(F[i]*i!, i=1..10);
1, 2, 9, 64, 625, 7776, 117649, 2097152, 43046721, 1000000000
> 
> 
> #Attendence Q7:
> #How many labeled trees are there with 27 vertices and 6 leaves?
> 
> L:=TreeSeqL(27,t)[27];
L := t^26+872415206*t^25+275346987688500*t^24+2920731168361734000*t^23+4388194382312972322000*t^22+1772864567898955712781600*t^21+270505223936259661308960000*t^20+19061791647342486356714400000*t^19+705593937039936089475330000000*t^18+14966140358214161238778175520000*t^17+193312731215363516295842983680000*t^16+1588126521127294558253647964160000*t^15+8560829696851107798143961139200000*t^14+30950985756049703203067092684800000*t^13+76152817742685615910771752960000000*t^12+128573373297423908381148433612800000*t^11+149337400551313221224370647654400000*t^10+118944248416147713732633427968000000*t^9+64352984586434875296725760000000000*t^8+23255103121345343784210554880000000*t^7+5466338131409680923049107456000000*t^6+802984907071337618164432896000000*t^5+69335884454191673897779200000000*t^4+3189363305076239568076800000000*t^3+65534862433073415782400000000*t^2+403291461126605635584000000*t
> #5466338131409680923049107456000000
> 
> #Attendence Q8:
> #Conjecture an explicit formula for the average number of leaves of a labeled trees with n vertices.
> average27:=sum(coeff(L,t,i), i=1..27)/27; 
average27 := 22528399544939174411840147874772641
> #What is the limit if you divide by n
> #Prove the conjecture.
> 
> evalb(average27=27^24);
true
> Conjecture: Average number of leaves of a labeled trees with n vertices is n^(n-3)
> 
> 
>