Data Structures - Binary Trees and Operations on Binary Trees
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1 ata Structures - inary Trees and Operations on inary Trees MS 275 anko drovic (UI) MS 275 October 15, / 25
2 inary Trees binary tree is a finite set of elements. It can be empty partitioned into three disjoint subsets The structure of the three disjoint subsets is as follows: a single element, called the root of the tree subsets two and tree are themselves binary trees, called left subtree and right subtree that is, left and right subtrees of the original tree each element in a binary tree is called a node R igure: binary tree anko drovic (UI) MS 275 October 15, / 25
3 inary Trees binary tree has a specific structure each node can have at most two branches R igure: This is still a binary tree anko drovic (UI) MS 275 October 15, / 25
4 Not a inary Trees binary tree has a specific structure each node can have at most two branches R G igure: Not a binary tree: fails the binary condition anko drovic (UI) MS 275 October 15, / 25
5 Not a inary Trees binary tree has a specific structure each node can have at most two branches trees do not contain loops R igure: Not a binary tree: fails the tree condition. Trees do not contain loops! anko drovic (UI) MS 275 October 15, / 25
6 complete inary Tree nodes,,, are called leaves a strictly binary tree with n leaves has 2n 1 nodes R igure: complete binary tree anko drovic (UI) MS 275 October 15, / 25
7 (complete) inary Tree Nodes in a tree can be categorized by their level the root (R) has level 0 the nodes and have level 1 leaves (,,,) form the last level, in this case 2 the depth of a binary tree is the maximum of any leaf in the tree R igure: complete binary tree of depth 2 anko drovic (UI) MS 275 October 15, / 25
8 complete inary Tree If a binary tree contains m nodes at level l it contains at most 2m nodes at level l + 1 urthermore, a binary tree can contain at most 2 l nodes at level l the total number of nodes in a complete binary tree of depth d, denoted TN, equals the sum of nodes at each level between 0 and d d TN = d = 2 k R k=0 igure: complete binary tree of depth 2 with = 7 nodes. anko drovic (UI) MS 275 October 15, / 25
9 pplications of inary Trees a binary tree is a useful data structure for binary decision-making for example, finding duplicates in a list direct approach: comparison of each element with those preceeding it this number of comparisons can be very large using a binary binary tree reduces this number significantly The process is described in the following steps: 1 make the first number in the list the root node, with left and right subtrees empty 2 compare each successive number with the root 3 if it matches, we have a duplicate. Otherwise, we make comparisons with the root if it is smaller, we examine the left subtree if it is larger, we examine the right subtree 4 if the subtree is empty, the number is not a duplicate, and it is placed into a new node at that position in the tree 5 if the subtree is not empty, we compare the number to the contents of the root of the subtree, and the entire process is repeated with the subtree. anko drovic (UI) MS 275 October 15, / 25
10 inding uplicates with a inary Tree ind (and remove) duplicates from the list [14, 15, 4, 9, 7, 18, 3, 5, 16, 4, 20, 17, 9, 14, 5] Recall: smaller than root - left larger than root - right otherwise - duplicate repeat on subtree igure: uplicate-free elements of the list anko drovic (UI) MS 275 October 15, / 25
11 Operations on inary Trees: Traversal common operation on a binary tree is traversal. That is, nodes in the tree often contain some useful information traverse (pass) through the tree we may want to print the content of each node we may want to process the data contained in the nodes in some other way In case of lists: [1,2,43,23,4,56,6,4,54,43,17,4,2] traversal is straight forward - we traverse it in a linear fashion we loop through the list and visit each element we may want to start at a specific index and may end at specific index but in general we visit all elements from first to last within a specified range there is an obvious or natural way to traverse a list anko drovic (UI) MS 275 October 15, / 25
12 Operations on inary Trees: Traversal igure: Traversal of a binary tree is not obvious. or a tree or a binary tree: there is no natural or obvious way of how to traverse anko drovic (UI) MS 275 October 15, / 25
13 Operations on inary Trees: Traversal Since there are no natural ways to traverse a binary tree: we will define three methods of traversal traversal methods will be recursive many operations on binary trees are recursive The three methods of traversal are preorder (depth-first order) inorder (symmetric order) postorder anko drovic (UI) MS 275 October 15, / 25
14 Operations on inary Trees: Preorder Traversal Recursive traversal in preorder performs the following operations: 1 visits the root node (read/write) 2 traverses the left subtree in preorder (recursive call) 3 traverses the right subtree in preorder (recursive call) anko drovic (UI) MS 275 October 15, / 25
15 Operations on inary Trees: Inorder Traversal Recursive traversal in inorder performs the following operations: 1 traverses the left subtree in inorder (recursive call) 2 visits the root node (read/write) 3 traverses the right subtree in inorder (recursive call) anko drovic (UI) MS 275 October 15, / 25
16 Operations on inary Trees: Postorder Traversal Recursive traversal in postorder performs the following operations: 1 traverses the left subtree in postorder (recursive call) 2 traverses the right subtree in postorder (recursive call) 3 visits the root node (read/write) anko drovic (UI) MS 275 October 15, / 25
17 Operations on inary Trees: Preorder Traversal 1 visit the root node (read/write) 2 traverse the left subtree in preorder (recursive call) 3 traverse the right subtree in preorder (recursive call) G H I igure: preorder: GHI anko drovic (UI) MS 275 October 15, / 25
18 Operations on inary Trees: Inorder Traversal 1 traverse the left subtree in inorder (recursive call) 2 visit the root node (read/write) 3 traverse the right subtree in inorder (recursive call) G H I igure: inorder: GHI anko drovic (UI) MS 275 October 15, / 25
19 Operations on inary Trees: Postorder Traversal 1 traverse the left subtree in postorder (recursive call) 2 traverse the right subtree in postorder (recursive call) 3 visit the root node (read/write) G H I igure: postorder: GHI anko drovic (UI) MS 275 October 15, / 25
20 Operations on inary Trees: Preorder Traversal 1 visit the root node (read/write) 2 traverse the left subtree in preorder (recursive call) 3 traverse the right subtree in preorder (recursive call) G H I J K L igure: anko drovic (UI) MS 275 October 15, / 25
21 Operations on inary Trees: Preorder Traversal 1 visit the root node (read/write) 2 traverse the left subtree in preorder (recursive call) 3 traverse the right subtree in preorder (recursive call) G H I J K L igure: preorder: IJGHKL anko drovic (UI) MS 275 October 15, / 25
22 Operations on inary Trees: Inorder Traversal 1 traverse the left subtree in inorder (recursive call) 2 visit the root node (read/write) 3 traverse the right subtree in inorder (recursive call) G H I J K L igure: anko drovic (UI) MS 275 October 15, / 25
23 Operations on inary Trees: Inorder Traversal 1 traverse the left subtree in inorder (recursive call) 2 visit the root node (read/write) 3 traverse the right subtree in inorder (recursive call) G H I J K L igure: inorder: IJGKHL anko drovic (UI) MS 275 October 15, / 25
24 Operations on inary Trees: Postorder Traversal 1 traverse the left subtree in postorder (recursive call) 2 traverse the right subtree in postorder (recursive call) 3 visit the root node (read/write) G H I J K L igure: anko drovic (UI) MS 275 October 15, / 25
25 Operations on inary Trees: Postorder Traversal 1 traverse the left subtree in postorder (recursive call) 2 traverse the right subtree in postorder (recursive call) 3 visit the root node (read/write) G H I J K L igure: postorder: IJGKLH anko drovic (UI) MS 275 October 15, / 25
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