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From: Ben Bacarisse <ben@bsb.me.uk>
Newsgroups: sci.math
Subject: Re: WM and end segments...
Date: Mon, 22 Jul 2024 00:10:46 +0100
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"Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> writes:

> For some damn reason when I hear end segments from WM I think of a
> tree. Take the following infinite 2-ary tree that holds the positive
> integers:
> ___________________________________________
>                   0
>                  / \
>                 /   \
>                /     \
>               /       \
>              1         2
>             / \       / \
>            /   \     /   \
>           3     4   5     6
>          / \   / \ / \   / \
>       .........................
> ___________________________________________
>
> this goes on and on for infinity... We all can see how this can go for
> infinity, right WM? Wrt trees there are only leaves in a finite view of
> it. However, the "infinite view" of the tree has no leafs because it never
> ends... Fair enough? Or too out there?

That's a can of worms in WMaths.  WM has written 734,342,120 nonsense
posts about binary trees over the years.  It's one of his favourite
examples to use to bamboozle his poor students.

The infinite binary tree -- simply a graph with node set N and edge set
(n, 2n+2) (in your numbering) -- is a particular puzzle for WM because
the node and edge sets are countable but the path set isn't.

Can you see a proof that the infinite rooted paths can be mapped, one to
one, with an uncountable subset of R?

> ... The infinite one has no leaves.

If you consider graphs in general, they do not have to be infinite to
have no leaves.

-- 
Ben.