Tree Terminology, Terminable & Interminable

[Jon Awbrey]

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oh, i understand that dialects in graph theory differ,
especially when you have to interface with computer sci,
i am merely going by the particulat usage that i learned
long ago, and there's always a way of finding the proper
translations between different terminologies.

"planted plane tree" was a classical usage for the trees
that are counted by catalans.  different folks would use
different strokes like these to indicate a "planter":

o  OR  o  OR  o
^      |      |
       -      @

and sometimes they would count the root and sometimes not.
the point here is to prevent the rotations around the root
within the plane, not to block the reflections, which are
not allowed anyway.

e.g. "the five planted planes of order 5"

o
|
o      o   o        o    o
|       \ /         |    |
o        o      o   o    o   o    o o o
|        |       \ /      \ /      \|/
o        o        o        o        o
|        |        |        |        |
@        @        @        @        @

p.67 in Harary & Palmer, 'Graphical Enumeration', 1973.

i was in the middle of writing another note slighly related to this,
but i think that all of this fuss about the planters may have been
one of the reasons why i eventually shifted to cacti, which are
much happier without the planters, the desert sand being far
too forbidding of these underground rotations in any case.

jon awbrey

re:

AK: | The rooted plane trees encoded here are:
    | .....................o...............o.........o...o..o.......
    | .....................|...............|..........\./...|.......
    | .......o....o...o....o....o.o.o..o...o.o.o.o.o...o....o...o...
    | .......|.....\./.....|.....\|/....\./...\|.|/....|.....\./....
    | @...... at ......@...... at ......@...... at ......@...... at ......@.....
    | 1......2......3......4......5......6......7......8......9.....

JA: oh, strictly speaking, we should be calling these "planted plane trees"
    as otherwise you can rotate the branches around the root and transform,
    for example, 6 into 9.

what i meant is that we have to add a planter to the above trees
in order to get a correspondence with parenthetical expressions.

AK: I'm following Stanley's usage in his
    "Exercises on Catalan and Related Numbers"
    http://www-math.mit.edu/~rstan/ec/catalan.pdf

etc.

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