#Please do not post homework
#Ravali Bommanaboina, 12/6/20, Assignment 22
#Question 1
#How many labeled ROOTED trees are there with 67 vertices and 12 leaves?
#76266803559579100826176891805054624762686813130778915491630013984735728176683034501075435748170137600000000000000
#How many labeled connected graphs are there with 40 vertices and 43 edges?
#90324445150366623501655158316607196285055246080819484164096000000000

#Question 2
SeqRTchild:=proc(S,N) local i,j:
i:=0;
j:=[];
for i in S do
j:=[op(j),coeff(taylor(TreeSeqL(N,t)[N],t=0,N+1),t,i)];
od;
end:
#S={1,2}
#[265252859812191058636308480000000, 53846330541874784903170621440000000]
#Not in OEIS
#S={1,2,3}
#[265252859812191058636308480000000, 53846330541874784903170621440000000, 3311549328325299271544993218560000000]
#Not in OEIS
#S={1,2,3,4}
#[265252859812191058636308480000000, 53846330541874784903170621440000000, 3311549328325299271544993218560000000, 92137802348465491317437829611520000000]
#Not in OEIS
#S={1,3,4}
#[265252859812191058636308480000000, 3311549328325299271544993218560000000, 92137802348465491317437829611520000000]
#Not in OEIS
#S={2,3,4}
#[53846330541874784903170621440000000, 3311549328325299271544993218560000000, 92137802348465491317437829611520000000]

#Question 3
SeqRTchildNone:=proc(S,N) local i,j,k:
i:=0;
j:={seq(i,i=1..N-1)} minus S;
k:=[];
for i in j do
k:=[op(k),coeff(taylor(TreeSeqL(N,t)[N],t=0,N+1),t,i)];
od;
end:

#S={1,2}
#[3311549328325299271544993218560000000, 92137802348465491317437829611520000000, 1384954544702290405101999968747520000000, 12429415786894133942680003510272000000000, 70937298255600522949958276736614400000000, 268562451967158853927617676760678400000000, 694351979139534343495428350150000640000000, 1250695680349629931963927834929315840000000, 1590397663985758919729999340207267840000000, 1438967744240806620911240026911774720000000, 929393507202873366099480794601633792000000, 428085219716023753298110491499511808000000, 139899516748656074187838134955905024000000, 32131310484298956675795213841117440000000, 5112144299969752570297565183194003200000, 552129577593540534119785376918995200000, 39379700403152019286149262394505600000, 1787180479507111057447754904228000000, 49077690146916782396301287583552000, 760906743356578800032865043776000, 6037954081855952080694373504000, 21216998851131035575245168000, 26338535975351791439604720, 7891430284748605820400, 278632793013729000, 233538845850, 30]
#Not in OEIS
#S={1,2,3}
#[92137802348465491317437829611520000000, 1384954544702290405101999968747520000000, 12429415786894133942680003510272000000000, 70937298255600522949958276736614400000000, 268562451967158853927617676760678400000000, 694351979139534343495428350150000640000000, 1250695680349629931963927834929315840000000, 1590397663985758919729999340207267840000000, 1438967744240806620911240026911774720000000, 929393507202873366099480794601633792000000, 428085219716023753298110491499511808000000, 139899516748656074187838134955905024000000, 32131310484298956675795213841117440000000, 5112144299969752570297565183194003200000, 552129577593540534119785376918995200000, 39379700403152019286149262394505600000, 1787180479507111057447754904228000000, 49077690146916782396301287583552000, 760906743356578800032865043776000, 6037954081855952080694373504000, 21216998851131035575245168000, 26338535975351791439604720, 7891430284748605820400, 278632793013729000, 233538845850, 30]
#Not in OEIS
#S={1,2,3,4}
#[1384954544702290405101999968747520000000, 12429415786894133942680003510272000000000, 70937298255600522949958276736614400000000, 268562451967158853927617676760678400000000, 694351979139534343495428350150000640000000, 1250695680349629931963927834929315840000000, 1590397663985758919729999340207267840000000, 1438967744240806620911240026911774720000000, 929393507202873366099480794601633792000000, 428085219716023753298110491499511808000000, 139899516748656074187838134955905024000000, 32131310484298956675795213841117440000000, 5112144299969752570297565183194003200000, 552129577593540534119785376918995200000, 39379700403152019286149262394505600000, 1787180479507111057447754904228000000, 49077690146916782396301287583552000, 760906743356578800032865043776000, 6037954081855952080694373504000, 21216998851131035575245168000, 26338535975351791439604720, 7891430284748605820400, 278632793013729000, 233538845850, 30]
#Not in OEIS
#S={1,3,4}
#[53846330541874784903170621440000000, 1384954544702290405101999968747520000000, 12429415786894133942680003510272000000000, 70937298255600522949958276736614400000000, 268562451967158853927617676760678400000000, 694351979139534343495428350150000640000000, 1250695680349629931963927834929315840000000, 1590397663985758919729999340207267840000000, 1438967744240806620911240026911774720000000, 929393507202873366099480794601633792000000, 428085219716023753298110491499511808000000, 139899516748656074187838134955905024000000, 32131310484298956675795213841117440000000, 5112144299969752570297565183194003200000, 552129577593540534119785376918995200000, 39379700403152019286149262394505600000, 1787180479507111057447754904228000000, 49077690146916782396301287583552000, 760906743356578800032865043776000, 6037954081855952080694373504000, 21216998851131035575245168000, 26338535975351791439604720, 7891430284748605820400, 278632793013729000, 233538845850, 30]
#Not in OEIS
#S={2,3,4}
#[265252859812191058636308480000000, 1384954544702290405101999968747520000000, 12429415786894133942680003510272000000000, 70937298255600522949958276736614400000000, 268562451967158853927617676760678400000000, 694351979139534343495428350150000640000000, 1250695680349629931963927834929315840000000, 1590397663985758919729999340207267840000000, 1438967744240806620911240026911774720000000, 929393507202873366099480794601633792000000, 428085219716023753298110491499511808000000, 139899516748656074187838134955905024000000, 32131310484298956675795213841117440000000, 5112144299969752570297565183194003200000, 552129577593540534119785376918995200000, 39379700403152019286149262394505600000, 1787180479507111057447754904228000000, 49077690146916782396301287583552000, 760906743356578800032865043776000, 6037954081855952080694373504000, 21216998851131035575245168000, 26338535975351791439604720, 7891430284748605820400, 278632793013729000, 233538845850, 30]
#Not in OEIS

#Question 4
#Estimate the limit of the average number of leaves in a labelled tree with vertex n divided by n
#0.372
#Estimate the limit of the standard-deviation of the number of the random variable ` number of leaves in a labelled tree with vertex n', divided by n
#K=6
#0.49
#K=3
#0.49
#K=4
#0.49
#K=5
#0.49