CS301-Lec27 handout - CS301 Data Structures Lecture No. 27...

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CS301 – Data Structures Lecture No. 27 ___________________________________________________________________ Data Structures Lecture No. 27 Reading Material Data Structures and Algorithm Analysis in C++ Chapter. 4 4.3 Summary Properties of Binary Tree Threaded Binary Trees Adding Threads During Insert Where is Inorder Successor? Inorder Traversal Properties of Binary Tree By the end of the last lecture, we were having a discussion about the properties of the binary trees. Let us recall, that I told you about a property of the binary trees regarding relationship between internal nodes and external nodes i.e. If the number of internal nodes is N, the number of external nodes will be N+1. Today I am going to discuss another property of the binary trees, which together with the previous lecture, will give us a start into a new topic. Let me have your attention to the second property of the binary trees. Property A binary tree with N internal nodes has 2N links, N-1 links to internal nodes and N+1 links to external nodes. Please recall that the first property dealt with the relationship between internal and external nodes. This property is dealing with the relationship of links to the internal nodes. Now, what is a link? As you might already have understood, a link is that line, which we draw between two nodes in a tree. Internally we use pointers in C++ to realize links. In pictorial sketches, however, we use a line to show a link between the two nodes. The property defines, that if you have a binary tree with Nodes, how many links, it will have between the internal nodes (remember, it includes the leaf nodes), and how many links it will have between the external nodes. We have not been showing any links between the external nodes in the diagrams. These are, in fact, null pointers. That means, these are the links, which we will show with the help of the square nodes. Let us see a binary tree, on the basis of which, we will further explore this property. In the following figure, the Page 1 of 14
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CS301 – Data Structures Lecture No. 27 ___________________________________________________________________ binary tree is shown again, which, in addition to the normal links between the internal nodes, also contains external nodes as squares and the external links as lines going to those squares. A B C D E F G E F Internal link external link Internal links: 8 External links: 10 Fig 27.1 Now if you count the total number of links in the diagram between internal and external nodes, it will be 2N. Remember, we are talking about links and not nodes. In this tree, we have 9 nodes marked with capital letters, 8 internal links and 10 external links. Adding the both kinds of links, we get 18, which is exactly 2 x 9. As discussed already that these properties are mathematical theorems and can therefore
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CS301-Lec27 handout - CS301 Data Structures Lecture No. 27...

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