Unit06A - 1 1 Trees Binary Trees 6A 2 Trees 3 Binary Trees...

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Unformatted text preview: 1 1 Trees Binary Trees 6A 2 Trees 3 Binary Trees A binary tree is either empty or it contains a root node and left- and right-subtrees that are also binary trees. A binary tree is a nonlinear data structure. The top node of a tree is called the root . Any node in a binary tree has at most 2 children . Any node in a binary tree has exactly one parent node (except the root). 4 Tree Terminology A G F E D C B root parent left-child right-child leaf left-subtree right-subtree 5 Types of Binary Trees Expression Trees * + - / 2 5 6 3 7 (6 / 2 + 5) * (7 - 3) 6 Types of Binary Trees Huffman Trees C D x B x y y A z 1 1 1 A 45% B 30% C 20% D 5% A 1 B 00 C 010 D 011 1001010100 = ABACAB 2 7 Types of Binary Trees Binary Search Trees 84 41 96 24 37 50 13 98 8 More Terminology A full binary tree is a binary tree such that - all leaves have the same level, and - every non-leaf node has 2 children. A complete binary tree is a binary tree such that - every level of the tree has the maximum number of nodes possible except possibly the deepest level, and - at the deepest level, the nodes are as far left as possible. 9 Examples A G F E D C B A E D C B FULL (and COMPLETE) COMPLETE 10 Binary Trees & Recursion Consider two nodes in a tree, X and Y. X is an ancestor of Y if X is the parent of Y, or X is the ancestor of the parent of Y. It’s RECURSIVE! X is a descendant of Y if X is the child of Y, or X is the descendant of the child of Y. 11 Binary Trees Levels A G F E D C B Level 1 Level 2 Level 3 12 Binary Trees - Levels The level of a node Y is BASE CASE RECURSIVE CASE 1, if the Y is the root 1 + the level of the parent of Y, if Y is not the root 3 13 Binary Tree - Height The height of a binary tree T is the number of nodes in the longest path from the root node to a leaf node. BASE CASE RECURSIVE CASE 0, if T is empty if T is not empty, max(height(left(T)), height(right(T))) + 1 14 Binary Tree Traversals A G F C D E B PREORDER ABDECFG INORDER DBEAFCG POSTORDER DEBFGCA 15 Traversal Example A G E D C B preorder inorder postorder F H I ABDFGCEHI BFDGAEIHC FGDBIHECA 16 Traversals are Recursive Preorder traversal 1. Visit the root....
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Unit06A - 1 1 Trees Binary Trees 6A 2 Trees 3 Binary Trees...

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