The Forest Primeval

[Jon Awbrey]

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

Here is what the initial mapping of natural numbers
into parenthetical expressions looked like:

1     2^0                          o           ()
2     2^2^0                        (o)         (())
3     2^0 3^2^0                    o(o)        ()(())
4     2^2^2^0                      ((o))       ((()))
5     2^0 3^0 5^2^0                oo(o)       ()()(())
6     2^2^0 3^2^0                  (o)(o)      (())(())
7     2^0 3^0 5^0 7^2^0            ooo(o)      ()()()(())
8     2^[2^0 3^2^0]                (o(o))      (()(()))
9     2^0 3^2^2^0                  o((o))      ()((()))
10    2^2^0 3^0 5^2^0              (o)o(o)     (())()(())
11    2^0 3^0 5^0 7^0 11^2^0       oooo(o)     ()()()()(())
12    2^2^2^0 3^2^0                ((o))(o)    ((()))(())
13    2^0 3^0 5^0 7^0 11^0 13^0    ooooo(o)    ()()()()()(())
14    2^2^0 3^0 5^0 7^2^0          (o)oo(o)    (())()()(())
15    2^0 3^2^0 5^2^0              o(o)(o)     ()(())(())
16    2^2^2^2^0                    (((o)))     (((())))

The "o" is just an easier to read variant
for an empty pair of terminal parentheses.

Here is the second half of that note I wrote,
where I "supposedly" got a 1-1 correspondence:

| The correspondence becomes one-to-one by the following transformation.
| To the rightmost "offspring", when not the only offspring, of any node
| of a given planted plane tree, assign an extra unit of significance:
| such that, if terminal, that node represents a unit exponent;  if it
| is not terminal, the unit may be "inherited", but only by a singleton
| offspring of that node, with a like condition entailing any further
| inheritance, until a terminal node reached;  otherwise, the significance
| is lost.  E.g., 360 is now uniquely represented by the following tree,
| where primed nodes count for unit exponents under the above rule:
|
|           o
|           |
|      o o' o
|       \|  |
|        o  o  o'
|         \ | /
|          \|/
|           o
|           |
| 360   =   @

Subsequently, I wrote this marginal note:

| e.g., now:
|
|              o            o   o
|              |             \ /
| o            o              o
| |            |              |
| @ = () = 1,  @ = (()) = 2,  @ = ()() = 3

Well, that is what I wrote, but to be honest it looks wrong somehow.
Maybe I muffed up the initial conditions in my later note.  Or maybe
it can be fixed by deleting the "when not the only offspring" clause
in the token-passing transformation.  Let me give it another try ...

What we are really interested in are sequences of exponents,
that we turn into sequences of trees, and we use the gimmick
of a graph-theoretical "planter" to convert the sequences of
trees into planted plane tres.  In the general setting, the
planter look like this:

^ ^ ^
 \|/
  o
  |
  @

But it's a quirky property of this set-up that the
empty sequence and the singleton sequences require
planters that appear to be excessively large:

Empty sequence <>:

o
|                 
@

Singleton sequence <k>:

k
|
o
|
@

So maybe I should have written:

|              o'           o   o'
|              |             \ /
| o            o              o
| |            |              |
| @ = _ = 1,   @ = () = 2,    @ = ()() = 3

So far this seems to suggest that the buck-passing rule
should apply to every node but the root, but now I will
have to rethink the whole thing.  Maybe I'll just leave
this one to the current inhabitants of Fair Catalania,
and return to the Realm Of Rote Arithmetic.

Jon Awbrey

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o