Question 1:  How many labeled ROOTED trees are there with 67 vertices and 12 leaves? (recall that a leaf is a vertex with no children) (ii) How many labeled connected graphs are there with 40 vertices and 43 edges?

Answer 1: 
We use the procedure TreeSeqL(N, t)
i) For Rooted trees with 67 vertices and 12 leaves:
762668035595791008261768918050546247626868131307789154916300139847357281766830345010754357481701376000000000000000

ii) You use the command ATreeSeq(40, 4)[40]
Ans
90324445150366623501655158316607196285055246080819484164096000000000

Question 2:
Write a procedure SeqRTchild(S,N), that inputs a finite set of positive integers, S, and a positive integer N, and outputs the first N entries in the lhe list enumerating rooted labeled trees where every vertex is either a leaf (i.e. 0 children) or has a mumber of children that must be in the set S.
Output SeqRTchild(S,30) for the following sets S if it is in the OEIS, state the A-number. If it is not, say so.

(i) S={1,2}

(ii) S={1,2,3}

(iii) S={1,2,3,4}

(iv) S={1,3,4}

(v) S={2,3,4}

Question 3
Write a procedure SeqRTchildNone(S,N), that inputs a finite set of positive integers, S, and a positive integer N, and outputs the first N entries in lhe list enumerating rooted labeled trees where every vertex is either a leaf (i.e. 0 children) or has a mumber of children that must NOT be in the set S (otherwise it can have any number of children)
Output SeqRTchildNone(S,30) for the following sets S if it is in the OEIS, state the A-number. If it is not, say so.

(i) S={1,2}

(ii) S={1,2,3}

(iii) S={1,2,3,4}

(iv) S={1,3,4}

(v) S={2,3,4}

Question 4:
By using procedure TreeSeqL(N,x) and procedure AveAndMoms(f,x,K) with K=6, added after class, try to
(i) Estimate the limit of the average number of leaves in a labelled tree with vertex n divided by n

(ii) 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

The third, fourth, fifth, and sixth scaled moments

Answer 4
i) It is 0.37
ii) This is the sequence i got for standard deviationL := [0.1571348402, 0.1457402977, 0.1335209347, 0.1233425370, 0.1149906649, 0.1080523762, 0.1021956270, 0.09717699729, 0.09281933018, 0.08899205425, 0.08559723085, 0.08256006771, 0.07982247273, 0.07733861994, 0.07507184788, 0.07299245333, 0.07107609347]
iii)[0.2357022603, 0.04621174720, 0.02131360716, 0.01304227277, 0.009162479703, 0.006963795701, 0.005562753972, 0.004597233260, 0.003894018938, 0.003360520378, 0.002942943349, 0.002607954534, 0.002333818015, 0.002105763504, 0.001913407645, 0.001749241095, 0.001607700989]
iv)[0.5000000000, 0.5944312328, 0.5353725782, 0.4613071085, 0.4021062583, 0.3555393622, 0.3183399688, 0.2880591791, 0.2629763365, 0.2418785886, 0.2238956692, 0.2083901969, 0.1948859865, 0.1830207124, 0.1725140824, 0.1631460113, 0.1547413581]
v)[0.5892556507, 0.6962569910, 0.2400269516, 0.1343595885, 0.09182986553, 0.06905021615, 0.05491477323, 0.04530490436, 0.03835613249, 0.03310465408, 0.02900237991, 0.02571444514, 0.02302454274, 0.02078661791, 0.01889843643, 0.01728627474, 0.01589562274]