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Trees
Chapter 4
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Objectives
Understand the terminology of the tree data structure Represent a tree structure in a program Understand the importance of the binary trees Use a binary search tree for storing ordered elements
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Trees Chapter 4 1 Objectives Understand the terminology of the - - PowerPoint PPT Presentation
Trees Chapter 4 1 Objectives Understand the terminology of the - - PowerPoint PPT Presentation
Trees Chapter 4 1 Objectives Understand the terminology of the tree data structure Represent a tree structure in a program Understand the importance of the binary trees Use a binary search tree for storing ordered elements 2 Motivation
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Motivation
Why lists, stacks, and queues are not enough? Not everything can be linearized. We may need to represent hierarchies, for example. Sorted array search: O(log(n)) Sorted array insert: O(n) Linked list search: O(n) Linked list insert: O(1) Can we build a data structure that is fast for both search and insert?
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Hierarchical Structures
UC System UCR BCOE CSE ECE CNAS … UCI UCSD …
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Hierarchical Structures
US CA Riverside County Riverside Palm Springs San Bernardino County … AZ MN
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…
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Definition
A tree can be defined recursively A tree is a group of nodes Each node contains a value If the tree is not empty, one node is identified as the root node The root node has zero or more subtrees The root of a subtree is connected to the root
- f the tree
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Terminology: Basic Definitions
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A B C D E I J F G H K Root Subtrees A is the parent of D D is the child of A B, C, and D are siblings E and F are not siblings
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Terminology: Descendants
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A B C D E I J F G H K Descendants of A
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Terminology: Ancestors
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A B C D E I J F G H K Ancestors of E Descendant of E
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Terminology: Leaves
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A B C D E I J F G H K Leaf nodes (Leaves) Internal nodes
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Terminology: Levels, Depth
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A B C D E I J F G H K Level 0 Level 1 Level 2 Level 3 J is at level 3 The depth of J is 3 What is the relationship between the depth of a node and the number of ancestors? What is the height of the tree?
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Terminology: Path
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A B C D E I J F G H K The path from A to J is (A, B, E, J) The length of the path is three (edges) What is the path from D to K? Is there a path from B to C?
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Tree Representation
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Node Value (any type) Children * * * * * template <type T> class Tree { class Node { T value; list<Node*> children; }; Node* root; };
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Parent Representation
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A B C D E I J F G H K template <type T> class Tree { class Node { T value; Node* parent; }; list<Node*> nodes; };
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Left-child Right-sibling
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A B C D E I J F G H K
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Left-child Right-sibling
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A B C D E I J F G H K template <type T> class Tree { class Node { T value; Node* left_child; Node* right_sibling; }; Node* root; };
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Binary Trees
A special case where every node has at most two children Has many applications that make it particularly interesting More restricted Room for optimization
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template <type T> class Tree { class Node { T value; Node* left; Node* right; }; Node* root; };
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Application: Expression Tree
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⨉ 3 5 / 4 2 ⨉ 2 +
3 × 5 + 4/2 × 2
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Inorder Tree Traversal
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⨉ 3 5 / 4 2 ⨉ 2 +
(3 × 5) + (4/2 ) × 2
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Postorder Tree Traversal
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⨉ 3 5 / 4 2 ⨉ 2 +
35 × 42/+2 ×
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Preorder Tree Traversal
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⨉ 3 5 / 4 2 ⨉ 2 +
×+×35/422
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Implementation of Traversals
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preorder(Node* root) { if (root == null) return; print(root->value); preorder(root->left); preorder(root->right); } postorder(Node* root) { if (root == null) return; postorder(root->left); postorder(root->right); print(root->value); } inorder(Node* root) { if (root == null) return; inorder(root->left); print(root->value); inorder(root->right); }