In article <firstname.lastname@example.org>, Zuhair <email@example.com> wrote:
> On Dec 13, 9:56 am, WM <mueck...@rz.fh-augsburg.de> wrote: > > > Ah I see, so you are imposing another condition on the definition of a > > > path, > > > > No, that is *the* definition of a path in a Binary Tree. > > > Actually it is not. There is no need at all to stipulate that a path > must begin by 0. It is a fixed > definition. > > I already showed you that the number of paths in a second degree > binary tree (or third degree if you want to adop the empty path) IS > larger than the total number of nodes.
For finite trees of more than one node there are always more nodes than paths.
And what I mean by paths those > that can start with 1 or with 0, but with the condition that it must > be unidirectional. And showed it clearly and I've illustrated each > path. You have 9 paths (inclusive of the empty path) and only 7 nodes. > > > > that it must begin with 0 since it is representing a number in > > > the interval [0,1], that's OK, then according to that I can see that > > > you are correct since the number of paths do corresponds to the number > > > of ending node, and with my calculation it would be less than the > > > total number of nodes by one. I'll need to check this again, but let > > > me agree with you on that for the current moment. > > > > > But still you have a problem. What you really managed to prove is that > > > the total number of FINITE undirectional paths beginning from 0 in the > > > infinite binary tree is countable, since we can simply draw an > > > injective map from those paths to their ending nodes. > > > > No. I proved that the number of infinite paths is countable by > > constructing all nodes of the Binbary Tree by a countable set of > > infinite paths.
That only shows that in a complete infinite binary treethe set of paths surjects onto the set of nodes, which does not prove what WM claims it proves. > > > > This only means that you can have a bijective function from a > countable subset of infinite paths of the binary tree to the set of > all nodes, which everyone already know that this is possible, because > we all agree that the total number of nodes of the infinite binary > tree is countable. > > What would be a proof is if you manage to define an injection from the > set of ALL infinite paths of the binary tree to the set of all nodes > of the binary tree. > > If you managed to do that, the next question is: > > where is that proof? please show us
Note that WM's attempt at a such a proof above was, as usual, trivially flawed > > Zuhair --