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| The \emph{complete infinite binary tree} consists of nodes (namely the numerals 0 and 1). Every node has two children which are not children of any other node. The tree serves as binary representation of all real numbers of the interval [0, 1] in form of paths, i.e., sequences of nodes. |
The \emph{complete infinite binary tree} consists of nodes (namely the numerals 0 and 1). Every node has two children which are not children of any other node. The tree serves as binary representation of all real numbers of the interval [0, 1] in form of paths, i.e., sequences of nodes. |
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| Every finite binary tree with more than one level contains less paths than nodes. Up to level \emph{n} there are $2^{n}$ paths and $2^{n+1} - 1$ nodes. |
Every finite binary tree with more than one level contains less paths than nodes. Up to level \emph{n} there are $2^{n}$ paths and $2^{n+1} - 1$ nodes. |
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| Every finite binary tree can be represented as an ordered set of nodes, enumerated by natural numbers. The union of all finite binary trees is then identical with the infinite binary tree. While the set of nodes |
Every finite binary tree can be represented as an ordered set of nodes, enumerated by natural numbers. The union of all finite binary trees is then identical with the infinite binary tree. While the set of nodes |
| remains countable, however, the set of paths is uncountable by Cantor's theorem. |
remains countable, however, the set of paths is uncountable by Cantor's theorem. |
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| \textbf{Literature} W. M\"uckenheim: Die Mathematik des Unendlichen, Shaker-Verlag, Aachen 2006. |
\textbf{Literature} |
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W. M\"uckenheim: Die Mathematik des Unendlichen, Shaker-Verlag, Aachen 2006. |
