Binary Search Trees. Lecture 29 Section Robb T. Koether. Hampden-Sydney College. Fri, Apr 8, 2016
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1 Binary Search Trees Lecture 29 Section 19.2 Robb T. Koether Hampden-Sydney College Fri, Apr 8, 2016 Robb T. Koether (Hampden-Sydney College) Binary Search Trees Fri, Apr 8, / 40
2 1 Binary Search Trees Searching a BST Inserting into a BST Deleting from a BST Count-Balancing a BST 2 Assignment Robb T. Koether (Hampden-Sydney College) Binary Search Trees Fri, Apr 8, / 40
3 Outline 1 Binary Search Trees Searching a BST Inserting into a BST Deleting from a BST Count-Balancing a BST 2 Assignment Robb T. Koether (Hampden-Sydney College) Binary Search Trees Fri, Apr 8, / 40
4 Binary Search Trees Definition (Binary Search Tree) A binary search tree is a binary tree with the following properties. There is a total order relation on the members in the tree. At every node, every member of the left subtree is less than or equal to the node value. At every node, every member of the right subtree is greater than or equal to the node value. Robb T. Koether (Hampden-Sydney College) Binary Search Trees Fri, Apr 8, / 40
5 A Binary Search Tree A binary search tree Robb T. Koether (Hampden-Sydney College) Binary Search Trees Fri, Apr 8, / 40
6 Binary Search Tree Interface Mutators void insert(const T& value); void remove(const T& value); insert() Insert a new node containing the value into the binary search tree. remove() Remove the node containing the value from the binary search tree. Robb T. Koether (Hampden-Sydney College) Binary Search Trees Fri, Apr 8, / 40
7 Binary Search Tree Interface Other Member Functions T* search(const T& value) const; void countbalance(); search() Search the binary search tree for the value. Return a pointer to the node where the value is found. Return NULL if the value is not found. countbalance() Count-balance the binary search tree. Robb T. Koether (Hampden-Sydney College) Binary Search Trees Fri, Apr 8, / 40
8 Balanced vs. Unbalanced Trees For the unbalanced tree, what is the average number of comparison? Robb T. Koether (Hampden-Sydney College) Binary Search Trees Fri, Apr 8, / 40
9 Balanced vs. Unbalanced Trees For the balanced tree, what is the average number of comparison? Robb T. Koether (Hampden-Sydney College) Binary Search Trees Fri, Apr 8, / 40
10 Outline 1 Binary Search Trees Searching a BST Inserting into a BST Deleting from a BST Count-Balancing a BST 2 Assignment Robb T. Koether (Hampden-Sydney College) Binary Search Trees Fri, Apr 8, / 40
11 Searching a BinarySearchTree Searching a Binary Search Tree Beginning at the root node, apply the following steps recursively. Compare the value to the node data. If it is equal, you are done. If it is less, search the left subtree. If it is greater, search the right subtree. If the subtree is empty, the value is not in the tree. Robb T. Koether (Hampden-Sydney College) Binary Search Trees Fri, Apr 8, / 40
12 Outline 1 Binary Search Trees Searching a BST Inserting into a BST Deleting from a BST Count-Balancing a BST 2 Assignment Robb T. Koether (Hampden-Sydney College) Binary Search Trees Fri, Apr 8, / 40
13 Inserting a Value into a BinarySearchTree Inserting into a Binary Search Tree Beginning at the root node, apply the following steps recursively. Compare the value to the node data. If it is less (or equal), continue recursively with the left subtree. If is is greater, continue recursively with the right subtree. When the subtree is empty, attach the node as a subtree. Robb T. Koether (Hampden-Sydney College) Binary Search Trees Fri, Apr 8, / 40
14 Outline 1 Binary Search Trees Searching a BST Inserting into a BST Deleting from a BST Count-Balancing a BST 2 Assignment Robb T. Koether (Hampden-Sydney College) Binary Search Trees Fri, Apr 8, / 40
15 Deleting a Value from a BinarySearchTree Perform a search to locate the value. This node will have Two children, or One child, or No child. Robb T. Koether (Hampden-Sydney College) Binary Search Trees Fri, Apr 8, / 40
16 A Binary Search Tree Robb T. Koether (Hampden-Sydney College) Binary Search Trees Fri, Apr 8, / 40
17 A Binary Search Tree Robb T. Koether (Hampden-Sydney College) Binary Search Trees Fri, Apr 8, / 40
18 Deleting a Value from a BinarySearchTree Case 1: No Child Delete the node. Robb T. Koether (Hampden-Sydney College) Binary Search Trees Fri, Apr 8, / 40
19 Delete a Node with No Child Robb T. Koether (Hampden-Sydney College) Binary Search Trees Fri, Apr 8, / 40
20 Delete a Node with No Child Robb T. Koether (Hampden-Sydney College) Binary Search Trees Fri, Apr 8, / 40
21 Deleting a Value from a BinarySearchTree Case 2: One Child Replace the node with the subtree of which the child is the root. Robb T. Koether (Hampden-Sydney College) Binary Search Trees Fri, Apr 8, / 40
22 Delete a Node with Two Children Robb T. Koether (Hampden-Sydney College) Binary Search Trees Fri, Apr 8, / 40
23 Delete a Node with Two Children Robb T. Koether (Hampden-Sydney College) Binary Search Trees Fri, Apr 8, / 40
24 Delete a Node with Two Children Robb T. Koether (Hampden-Sydney College) Binary Search Trees Fri, Apr 8, / 40
25 Deleting a Value from a BinarySearchTree Case 3: Two Children Locate the next smaller value in the tree. This value is the rightmost value of the left subtree. Move left one step. Move right as far as possible. Swap this value with the value to be deleted. The node to be deleted now has at most one child. Robb T. Koether (Hampden-Sydney College) Binary Search Trees Fri, Apr 8, / 40
26 Delete a Node with Three Children Robb T. Koether (Hampden-Sydney College) Binary Search Trees Fri, Apr 8, / 40
27 Delete a Node with Three Children Robb T. Koether (Hampden-Sydney College) Binary Search Trees Fri, Apr 8, / 40
28 Delete a Node with Three Children Robb T. Koether (Hampden-Sydney College) Binary Search Trees Fri, Apr 8, / 40
29 Delete a Node with Three Children Robb T. Koether (Hampden-Sydney College) Binary Search Trees Fri, Apr 8, / 40
30 Delete a Node with Three Children Robb T. Koether (Hampden-Sydney College) Binary Search Trees Fri, Apr 8, / 40
31 Outline 1 Binary Search Trees Searching a BST Inserting into a BST Deleting from a BST Count-Balancing a BST 2 Assignment Robb T. Koether (Hampden-Sydney College) Binary Search Trees Fri, Apr 8, / 40
32 Count-Balancing a BinarySearchTree Write a function movenoderight() that will move the largest value of the left subtree to the right subtree. The movenoderight() Function Locate the largest value in the left subtree. Delete it (but save the value). Place it at the root. Insert the old root value into the right subtree. Robb T. Koether (Hampden-Sydney College) Binary Search Trees Fri, Apr 8, / 40
33 A Binary Search Tree Robb T. Koether (Hampden-Sydney College) Binary Search Trees Fri, Apr 8, / 40
34 A Binary Search Tree Robb T. Koether (Hampden-Sydney College) Binary Search Trees Fri, Apr 8, / 40
35 A Binary Search Tree Robb T. Koether (Hampden-Sydney College) Binary Search Trees Fri, Apr 8, / 40
36 A Binary Search Tree Robb T. Koether (Hampden-Sydney College) Binary Search Trees Fri, Apr 8, / 40
37 Count-Balancing a BinarySearchTree Count-Balancing a Tree Write a similar function movenodeleft(). Apply either movenoderight() or movenodeleft() repeatedly at the root node until the tree is balanced at the root. Then apply these functions recursively, down to the leaves. Robb T. Koether (Hampden-Sydney College) Binary Search Trees Fri, Apr 8, / 40
38 Building a Binary Search Tree Suppose we wish to transmit the nodes of a balanced binary search tree to another computer and reconstruct the tree there. In what order should the values be transmitted? Robb T. Koether (Hampden-Sydney College) Binary Search Trees Fri, Apr 8, / 40
39 Building a Binary Search Tree We could use an in-order traversal to transmit them. At the receiving end, simply call insert() to insert each value into the tree. The constructed tree will be identical to the original. What do we get if we transmit the values using a pre-order traversal? Using a post-order traversal? Robb T. Koether (Hampden-Sydney College) Binary Search Trees Fri, Apr 8, / 40
40 Outline 1 Binary Search Trees Searching a BST Inserting into a BST Deleting from a BST Count-Balancing a BST 2 Assignment Robb T. Koether (Hampden-Sydney College) Binary Search Trees Fri, Apr 8, / 40
41 Assignment Assignment Read Section Robb T. Koether (Hampden-Sydney College) Binary Search Trees Fri, Apr 8, / 40
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