Help:=proc(): print(`BT(n), Images(T), UBT(n) `): end:

#BT(n): the set of rooted binary trees with n leaves
BT:=proc(n) local S,k,S1,S2,s1,s2:
option remember:
if n=1 then
  RETURN({[]}):
fi:

S:={}:

for k from 1 to n-1 do
 S1:=BT(k):
 S2:=BT(n-k):
S:= S union {seq( seq([s1,s2],s1 in S1), s2 in S2)}:
od:
S:
end:

#Images(T):
Images:=proc(T) local T1,T2,t1,t2,S:
if T=[] then
 RETURN({[]}):
fi:
T1:=Images(T[1]):
T2:=Images(T[2]):

S:={}:

for t1 in T1 do
 for t2 in T2 do
 S:=S union {[t1,t2],[t2,t1]}:
 od:
od:
S:
end:




#UBT(n): all the unlabeled binary trees with n vertices
UBT:=proc(n) local S1,S2,S3:
S1:=BT(n):

S2:={}:

while S1<>{} do
S3:=Images(S1[1]):
S2:=S2 union {S3[1]}:
S1:=S1 minus S3:
od:
S2:
end: