Descendants and ascendants in binary trees
There are three classical algorithms to visit all the nodes of a binary tree - preorder, inorder and postorder traversal. From this one gets a natural labelling of the n internal nodes of a binary tree by the numbers 1, 2, ..., n, indicating the sequence in which the nodes are visited. For given n (...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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Discrete Mathematics & Theoretical Computer Science
1997-01-01
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| Series: | Discrete Mathematics & Theoretical Computer Science |
| Subjects: | |
| Online Access: | https://dmtcs.episciences.org/246/pdf |
| _version_ | 1827323963332624384 |
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| author | Alois Panholzer Helmut Prodinger |
| author_facet | Alois Panholzer Helmut Prodinger |
| author_sort | Alois Panholzer |
| collection | DOAJ |
| description | There are three classical algorithms to visit all the nodes of a binary tree - preorder, inorder and postorder traversal. From this one gets a natural labelling of the n internal nodes of a binary tree by the numbers 1, 2, ..., n, indicating the sequence in which the nodes are visited. For given n (size of the tree) and j (a number between 1 and n), we consider the statistics number of ascendants of node j and number of descendants of node j. By appropriate trivariate generating functions, we are able to find explicit formulae for the expectation and the variance in all instances. The heavy computations that are necessary are facilitated by MAPLE and Zeilberger's algorithm. A similar problem comes fromlabelling the leaves from left to right by 1, 2, ..., n and considering the statistic number of ascendants (=height) of leaf j. For this, Kirschenhofer [1] has computed the average. With our approach, we are also able to get the variance. In the last section, a table with asymptotic equivalents is provided for the reader's convenience. |
| first_indexed | 2024-04-25T02:01:06Z |
| format | Article |
| id | doaj.art-d2840cd19366495ea60d2224396cb736 |
| institution | Directory Open Access Journal |
| issn | 1365-8050 |
| language | English |
| last_indexed | 2024-04-25T02:01:06Z |
| publishDate | 1997-01-01 |
| publisher | Discrete Mathematics & Theoretical Computer Science |
| record_format | Article |
| series | Discrete Mathematics & Theoretical Computer Science |
| spelling | doaj.art-d2840cd19366495ea60d2224396cb7362024-03-07T14:56:02ZengDiscrete Mathematics & Theoretical Computer ScienceDiscrete Mathematics & Theoretical Computer Science1365-80501997-01-01Vol. 110.46298/dmtcs.246246Descendants and ascendants in binary treesAlois Panholzer0Helmut Prodinger1Institut für Diskrete Mathematik und Geometrie [Wien]Institut für Diskrete Mathematik und Geometrie [Wien]There are three classical algorithms to visit all the nodes of a binary tree - preorder, inorder and postorder traversal. From this one gets a natural labelling of the n internal nodes of a binary tree by the numbers 1, 2, ..., n, indicating the sequence in which the nodes are visited. For given n (size of the tree) and j (a number between 1 and n), we consider the statistics number of ascendants of node j and number of descendants of node j. By appropriate trivariate generating functions, we are able to find explicit formulae for the expectation and the variance in all instances. The heavy computations that are necessary are facilitated by MAPLE and Zeilberger's algorithm. A similar problem comes fromlabelling the leaves from left to right by 1, 2, ..., n and considering the statistic number of ascendants (=height) of leaf j. For this, Kirschenhofer [1] has computed the average. With our approach, we are also able to get the variance. In the last section, a table with asymptotic equivalents is provided for the reader's convenience.https://dmtcs.episciences.org/246/pdfzeilberger's algorithmbinary treetree traversalgenerating function[info.info-dm] computer science [cs]/discrete mathematics [cs.dm] |
| spellingShingle | Alois Panholzer Helmut Prodinger Descendants and ascendants in binary trees Discrete Mathematics & Theoretical Computer Science zeilberger's algorithm binary tree tree traversal generating function [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] |
| title | Descendants and ascendants in binary trees |
| title_full | Descendants and ascendants in binary trees |
| title_fullStr | Descendants and ascendants in binary trees |
| title_full_unstemmed | Descendants and ascendants in binary trees |
| title_short | Descendants and ascendants in binary trees |
| title_sort | descendants and ascendants in binary trees |
| topic | zeilberger's algorithm binary tree tree traversal generating function [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] |
| url | https://dmtcs.episciences.org/246/pdf |
| work_keys_str_mv | AT aloispanholzer descendantsandascendantsinbinarytrees AT helmutprodinger descendantsandascendantsinbinarytrees |