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From: "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com>
Newsgroups: sci.math
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
 (extra-ordinary)
Date: Tue, 3 Dec 2024 13:05:25 -0800
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On 12/3/2024 12:29 AM, Moebius wrote:
> Am 03.12.2024 um 07:24 schrieb Moebius:
>> Am 03.12.2024 um 07:17 schrieb Chris M. Thomasson:
>>
>>>>>
>>>>>              0
>>>>>             / \
>>>>>            /   \
>>>>>           /     \
>>>>>          /       \
>>>>>         1         2
>>>>>        / \       / \
>>>>>       /   \     /   \
>>>>>      3     4   5     6
>>>>>   .........................
>>
>> Though we may take 1 for the root too. This way we would get (using 
>> binary representation):
>>
>>  >>>              1
>>  >>>             / \
>>  >>>            /   \
>>  >>>           /     \
>>  >>>          /       \
>>  >>>        10        11
>>  >>>        / \       / \
>>  >>>       /   \     /   \
>>  >>>     100   101 110   111
>>  >>>   .........................
>>
>> I guess you get the pattern. :-P
> 
> Another one (without numbers) but /left/ (l), /right/ (r):
> 
>>  >>>              *
>>  >>>             / \
>>  >>>            /   \
>>  >>>           /     \
>>  >>>          /       \
>>  >>>         l         r
>>  >>>        / \       / \
>>  >>>       /   \     /   \
>>  >>>     ll    lr   rl    rr
>>  >>>   .........................
> 
> This way we may even "identify" each node in the tree with a (finite) l- 
> r-sequence:
> 
>>  >>>             ()       [<<< the "empty l-r-sequence"]
>>  >>>            /  \
>>  >>>           /    \
>>  >>>          /      \
>>  >>>         /        \
>>  >>>       (l)        (r)
>>  >>>       / \        / \
>>  >>>      /   \      /   \
>>  >>>   (l,l) (l,r) (r,l) (r,r)
>>  >>>   .........................
> 
> :-P
> 
> So each node actually "is" (or represents) the path leading to it. :-P

(l, l) means take two left branches from the root.

(r, l) means take one right and one left from the root.

I see the pattern. It makes me think some more about my original tree 
with node, say, 6.

It's parent is (6-2)/2 = 2 that is a right (-2) wrt 6 has a parent at 2.

At node 2 take a right to get at node 6

Now, lets try 5. It's parent is (5-1)/2 = 2. That is a left from 2 wrt 
(-1). There is a pattern here as well.

At node 2 take a left to get at node 5.

Make any sense to you? Thanks.