Binary Trees Fall 2018 Margaret Reid-Miller

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1 Binary Trees Fall 2018 Margaret Reid-Miller

2 Trees Fall (Reid-Miller) 2

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). Fall (Reid-Miller) 3

4 Tree Terminology A root B C internal leaf D E F G Fall (Reid-Miller) 4

5 left-child Tree Terminology A parent B C right-child siblings D E F G Fall (Reid-Miller) 5

6 Tree Terminology left-subtree A root right-subtree B C D E F G Fall (Reid-Miller) 6

7 Types of Binary Trees Expression Trees * + - / (6 / 2 + 5) * (7-3) Fall (Reid-Miller) 7

8 Types of Binary Trees Huffman Trees coding A 45% B 30% C 20% D 5% 0 B 0 y 0 C z 1 x 1 A 1 D A 1 B 00 C 010 D = ABACAB Fall (Reid-Miller) 8

9 Types of Binary Trees Binary Search Trees Fall (Reid-Miller) 9

10 More Terminology A perfect binary tree is a binary tree such that - all leaves have the same level, and - every non-leaf node has 2 children. A full tree is a binary tree such that - 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. Fall (Reid-Miller) 10

11 Examples A A B C B C D E F G D E F PERFECT (and COMPLETE) COMPLETE (but not FULL) Fall (Reid-Miller) 11

12 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. Fall (Reid-Miller) 12

13 Binary Trees Levels A Level 0 B C Level 1 D E F G Level 2 Fall (Reid-Miller) 13

14 Binary Trees - Levels The level of a node Y is BASE CASE 0, if the Y is the root RECURSIVE CASE 1 + the level of the parent of Y, if Y is not the root Fall (Reid-Miller) 14

15 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 0, if T is empty RECURSIVE CASE max(height(left(t)), height(right(t))) + 1, if T is not empty Fall (Reid-Miller) 15

16 Binary Tree Traversals PREORDER INORDER POSTORDER A ABDECFG DBEAFCG DEBFGCA B C D E F G Fall (Reid-Miller) 16

17 Traversal Example B F D A G E C H preorder ABDFGCEHI inorder BFDGAEIHC postorder FGDBIHECA I Fall (Reid-Miller) 17

18 Traversals are Recursive Preorder traversal 1. Visit the root. 2. Perform a preorder traversal of the left subtree. 3. Perform a preorder traversal of the right subtree. Inorder traversal 1. Perform an inorder traversal of the left subtree. 2. Visit the root. 3. Perform an inorder traversal of the right subtree. Postorder traversal 1. Perform a postorder traversal of the left subtree. 2. Perform a postorder traversal of the right subtree. 3. Visit the root. Fall (Reid-Miller) 18

19 Traversals on Expression Trees What do you get when you perform a postorder traversal on an expression tree? + - * / / * Fall (Reid-Miller) 19

20 Traversals on Binary Search Trees What do you get when you perform an INORDER traversal on a binary search tree? Fall (Reid-Miller) 20

21 Implementing a binary tree Use an array to store the nodes. - mainly useful for complete binary trees (more on this soon) Use a variant of a linked list where each data element is stored in a node with links to the left and right children of that node Instead of a head reference, we will use a root reference to the root node of the tree. Fall (Reid-Miller) 21

22 Binary Tree Node (BTNode inner class) private class BTNode { private E data; private BTNode left; private BTNode right; } public BTNode(E element) { data = element; left = null; right = null; } data Fall (Reid-Miller) 22

23 BinaryTree class (Constructor) public class BinaryTree<E> { private BTNode root; public BinaryTree() { } root = null; Fall (Reid-Miller) 23

24 BinaryTree class (another constructor) public BinaryTree(E element, BinaryTree<E> lefttree, BinaryTree<E> righttree) { } root = new BTNode(element); if (lefttree!= null) root.left = lefttree.root; if (righttree!= null) root.right = righttree.root; Fall (Reid-Miller) 24

25 BinaryTree class short-circuit evaluation public BinaryTree<E> getleftsubtree() { } if (root!= null && root.left!= null) return new BinaryTree<E>(root.left); else return null; protected BinaryTree(BTNode rootref) { root = rootref; special constructor for this method only } same idea for right subtree Fall (Reid-Miller) 25

26 Preorder Traversal public String preorder() { return preorder(root); } private String preorder(node<e> node) { String result = ""; if (node!= null) { result += node.data + " "; result += preorder(node.left) + " "; result += preorder(node.right) + " "; } return result; } Fall (Reid-Miller) 26

CSE 230 Intermediate Programming in C and C++ Binary Tree

CSE 230 Intermediate Programming in C and C++ Binary Tree Fall 2017 Stony Brook University Instructor: Shebuti Rayana shebuti.rayana@stonybrook.edu Introduction to Tree Tree is a non-linear data structure