Binary Trees Page 1 Binary Trees A binary

Binary Trees Page 1 Binary Trees • • A binary tree is a linked structure that allows for a binary search strategy. A binary tree consists of a finite number of nodes (structured data objects). Each node can point to either 0, 1, or (at most) 2 different nodes (structures). Assume we have the following array of random integers: 9 12 7 14 10 4 3 8 13 If We wished to find an element on the list: • We could perform a sequential search (easy but slow) • We could sort and do a binary search (fast but difficult to sort and maintain) • A binary tree provides a compromise Data Structures in C for Non-Computer Science Majors Kirs and Pflughoeft

Binary Trees Page 2 Setting Up a Binary Tree Assume that each element on our list: 9 12 7 14 10 4 3 8 13 Is a node in our tree. The structure of each node will be: struct treetemplate { int value; struct treetemplate *left, *right; }; Take 1 element from the list (e. g. , the first one), and set it in as the first (root) node 9 Data Structures in C for Non-Computer Science Majors The first element is now the root of the tree Kirs and Pflughoeft

Binary Trees Page 3 Assume that this element were assigned location 12622 in RAM: 12622 - 12623 12624 -12627 12628 -12631 9 NULL Field: value Field: left Field: right Notice that the pointer fields (left and right) right initially point to nothing To insert the remaining elements from our array onto the list: list 1. Starting at the root, determine if the element is larger or smaller 2. Assume that larger elements will go the right, and smaller elements will go to the left. 3. Continue with the remaining elements on the list putting them in place in the tree. Data Structures in C for Non-Computer Science Majors Kirs and Pflughoeft

Binary Trees Page 4 Now add the next (second) element into the tree: 9 12 7 14 10 4 3 8 13 Since 12 is greater than 9, go to the right Root 9 If the second element (structure) were assigned to address 12720, 12720 RAM might appear as: 12 12622 - 12623 12624 -12627 12628 -12631 9 NULL 12720 - 12721 12722 -12725 12726 -12721 12 NULL Data Structures in C for Non-Computer Science Majors Kirs and Pflughoeft

Binary Trees Page 5 Now add the next (third) element into the tree: 9 12 14 7 Since 7 is less than 9, go to the left 10 4 3 8 13 9 If the third element were assigned to address 12730, 12730 RAM might appear as: 12 7 12622 - 12623 12624 -12627 12628 -12631 9 12730 12720 - 12721 12722 -12725 12726 -12729 12730 - 12731 12732 -12735 12736 -12739 12 NULL 7 NULL Data Structures in C for Non-Computer Science Majors Kirs and Pflughoeft

Binary Trees Page 6 Now add the next (fourth) element into the tree: 9 12 7 14 10 4 8 13 Since 14 is greater than 9, go to the right 9 7 If the fourth element were assigned to address 12756, 12756 RAM might appear as: 3 Since 14 is greater than 12, go right 12 14 12622 - 12623 12624 -12627 12628 -12631 9 12730 12720 - 12721 12722 -12725 12726 -12729 12730 - 12731 12732 -12735 12736 -12739 12 NULL 12756 7 NULL 12756 - 12757 12758 -12761 12762 -12765 14 NULL Data Structures in C for Non-Computer Science Majors Kirs and Pflughoeft

Binary Trees Page 7 Now add the next (fifth) element into the tree: 9 12 7 14 10 4 If the fifth element were assigned to address 12824, 12824 RAM might appear as: 12 10 8 13 Since 10 is greater than 9, go to the right Since 10 is less than 12, go left 9 7 3 14 12622 - 12623 12624 -12627 12628 -12631 9 12730 12720 - 12721 12722 -12725 12726 -12729 12730 - 12731 12732 -12735 12736 -12739 12 12824 12756 7 NULL 12756 - 12757 12758 -12761 12762 -12765 12824 - 12825 12826 -12829 12830 -12833 14 NULL 10 NULL Data Structures in C for Non-Computer Science Majors Kirs and Pflughoeft

Binary Trees Page 8 Adding the remaining elements onto the list: 9 12 7 14 10 4 3 8 13 9 7 4 3 Data Structures in C for Non-Computer Science Majors 12 8 10 14 13 Kirs and Pflughoeft

Binary Trees Page 9 RAM might appear as: 9 7 12622 -12623 12624 -12627 12628 -12631 9 12730 12720 12 4 8 14 10 3 13 12720 -12721 12722 -12725 12 12824 12756 -12757 12758 -12761 12762 -12765 12824 -12825 12826 -12829 12830 -12833 14 13310 NULL 12900 -12901 12902 -12905 4 12910 12726 -12729 12730 -12731 12756 12906 -12909 12910 -12911 NULL 13300 -12301 12302 -12305 8 NULL Data Structures in C for Non-Computer Science Majors 3 12736 -12739 12900 13300 7 12912 -12915 12916 -12919 NULL 12306 -12309 13310 -12311 NULL 12732 -12735 13 12312 -12315 12316 -12319 NULL Kirs and Pflughoeft

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Binary Trees Page 16 Binary Trees RECURSION • Recursion is a programming technique which allows a function to call itself. • Without recursion, traversing (moving about) binary trees becomes very difficult. Leaf Data Structures in C for Non-Computer Science Majors Kirs and Pflughoeft