[seqfan] Re: Definitions of "circular rooted trees, " "series-reduced mobiles, " "cycle rooted trees, " "asymmetric mobiles"

[franktaw at netscape.net]

Since no one else has answered, I'll give it a try. I'm fairly 
confident of these definitions based on looking at the sequences, but I 
didn't originate any of these terms, so I might be mistaken in some of 
the details.

I assume a basic understanding of graphs and trees, particularly rooted 
trees and ordered trees.

"Circular rooted trees" is synonymous with "mobiles"; I think "cycle 
rooted trees" is another synonym. A mobile is an intermediate case 
between a rooted tree and an ordered tree: the child nodes from a given 
parent can be rotated as a whole, but not permuted in other ways. For 
example (you'll want to view these with a fixed font),
...o...
../|\..
./.|.\.
o..o..o
...|..|
...|..|
...o..o
......|
......|
......o
is equivalent to
...o...
../|\..
./.|.\.
o..o..o
|..|...
|..|...
o..o...
...|...
...|...
...o...
since the subtrees are rotated left; but these are not equivalent to:
...o...
../|\..
./.|.\.
o..o..o
|..|...
|..|...
o..o...
|......
|......
o......
which differs from either by a non-rotational permutation.

"Series reduced", for any sort of rooted tree, generally means that no 
node has only a single child; there may be none, or two or more. (One 
can series reduce a rooted tree by repeatedly replacing any node with 
exactly one child by its subtree; the result will be independent of the 
order in which the replacements are made. Any of the 3 graphs shown 
above lead by series reduction to a tree whose root has three children, 
each a single node.) (Series reduced has a related but slightly 
different meaning for pure graphs, including free trees.)

Basically, asymmetric for a rooted tree means that there are no nodes 
with identical subtrees. This is a bit vague, because there is more 
than one possible definition, and I'm not sure what the author intends 
here.

Franklin T. Adams-Watters

-----Original Message-----
From: Thomas Copeland <tccopeland at gmail.com>

Dear seq. fans,

Where can I find good (preferably illustrated) definitions of the terms 
in
the subject line, which are used extensively in the OEIS?

Sincerely,

Tom Copeland