Data Structures and Algorithms for Information Processing Lecture

Binary Trees l A binary tree has nodes, similar to nodes in a linked

Binary Trees 90 -723: Data Structures and Algorithms for Information Processing Lecture 5: Trees

Some Terminology • Root is the unique node with no parent • Leaves or

Example use of Trees · Each internal node labeled by an operator · Each

Example Use of Trees Representing Chinese Characters • Characters composed hierarchically into boxes •

Binary vs. Binary-Unary Important distinction : Sometimes binary trees are defined to allow internal

Counting Trees The number of binary trees with internal nodes is There is no

Exercise Write efficient Java functions int num. BTrees(int n) int num. BUTrees(int n) that

Properties of Trees • There is exactly one path connecting any two nodes •

The Best Trees A binary tree is full if internal nodes fill every level,

Complete Binary Trees • A complete tree is one having and leaves levels •

Class for Binary Nodes public class BTNode { private Object data; private BTNode left;

Class for Binary Nodes public BTNode(Object obj, BTNode l, BTNode r) { data =

Copying Trees public static BTNode tree. Copy(BTNode t) { if (t == null) return

Tree Traversals P M E S A L R A T Preorder traversal 90

Tree Traversals P M L E S A R A T Inorder traversal 90

Tree Traversals P M E S A L R A T Postorder traversal 90

Tree Traversals P M E S A L R A T Level order traversal

Preorder Traversal 1 Process the root 2 Process the nodes in the left subtree

Preorder Print public void preorder. Print() { System. out. println(data); if (left != null)

Inorder Traversal 1 Profess the nodes in the left subtree 2 Process the root

Inorder Print public void preorder. Print() { if (left != null) left. preorder. Print();

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Data Structures and Algorithms for Information Processing Lecture 5: Trees 90 -723: Data Structures and Algorithms for Information Processing Lecture 5: Trees Copyright © 1999, Carnegie Mellon. All Rights Reserved.

Binary Trees l A binary tree has nodes, similar to nodes in a linked list structure. l Data of one sort or another may be stored at each node. l Each node is either a leaf, having no children, or an internal node , with one or two children 90 -723: Data Structures and Algorithms for Information Processing Lecture 5: Trees Copyright © 1999, Carnegie Mellon. All Rights Reserved.

Some Terminology • Root is the unique node with no parent • Leaves or terminals are nodes with no children • A subtreeis a node together with all its descendents • Level of a node is number of nodes on path from the node to root 90 -723: Data Structures and Algorithms for Information Processing Lecture 5: Trees Copyright © 1999, Carnegie Mellon. All Rights Reserved.

Example use of Trees · Each internal node labeled by an operator · Each leaf labeled by a variable or numeric value Arithmetic Expressions A*(((B+C)*(D*E))+F) ·Tree determines a unique value 90 -723: Data Structures and Algorithms for Information Processing Lecture 5: Trees Copyright © 1999, Carnegie Mellon. All Rights Reserved.

Example Use of Trees Representing Chinese Characters • Characters composed hierarchically into boxes • Boxes can be oriented left-right, top-down, or outside-inside • Each leaf labeled by one of 300 -400 radicals 90 -723: Data Structures and Algorithms for Information Processing Lecture 5: Trees Copyright © 1999, Carnegie Mellon. All Rights Reserved.

Binary vs. Binary-Unary Important distinction : Sometimes binary trees are defined to allow internal nodes with one or two children. There is a big difference. . . 90 -723: Data Structures and Algorithms for Information Processing Lecture 5: Trees Copyright © 1999, Carnegie Mellon. All Rights Reserved.

Counting Trees The number of binary trees with internal nodes is There is no such formula if we allow unary internal nodes! 90 -723: Data Structures and Algorithms for Information Processing Lecture 5: Trees Copyright © 1999, Carnegie Mellon. All Rights Reserved.

Exercise Write efficient Java functions int num. BTrees(int n) int num. BUTrees(int n) that return the number of binary trees and binary/unary trees, respectively. 90 -723: Data Structures and Algorithms for Information Processing Lecture 5: Trees Copyright © 1999, Carnegie Mellon. All Rights Reserved.

Properties of Trees • There is exactly one path connecting any two nodes • Any two nodes have a least common ancestor • A tree with N internal nodes has N+1 external nodes (easy to prove by induction) 90 -723: Data Structures and Algorithms for Information Processing Lecture 5: Trees Copyright © 1999, Carnegie Mellon. All Rights Reserved.

The Best Trees A binary tree is full if internal nodes fill every level, except possibly the last. The height of a full binary tree with internal nodes is 90 -723: Data Structures and Algorithms for Information Processing Lecture 5: Trees Copyright © 1999, Carnegie Mellon. All Rights Reserved.

Complete Binary Trees • A complete tree is one having and leaves levels • Complete trees have a simplementation using arrays (How? ) 90 -723: Data Structures and Algorithms for Information Processing Lecture 5: Trees Copyright © 1999, Carnegie Mellon. All Rights Reserved.

Class for Binary Nodes public class BTNode { private Object data; private BTNode left; private BTNode right; . . . 90 -723: Data Structures and Algorithms for Information Processing Lecture 5: Trees Copyright © 1999, Carnegie Mellon. All Rights Reserved.

Class for Binary Nodes public BTNode(Object obj, BTNode l, BTNode r) { data = obj; left = l; right= r; } public boolean is. Leaf() { return (left == null) && (right == null); }. . . 90 -723: Data Structures and Algorithms for Information Processing Lecture 5: Trees Copyright © 1999, Carnegie Mellon. All Rights Reserved.

Copying Trees public static BTNode tree. Copy(BTNode t) { if (t == null) return null; else { BTNode left. Copy = tree. Copy(t. left); BTNode right. Copy = tree. Copy(t. right); return new BTNode(t. data, left. Copy, right. Copy); } } 90 -723: Data Structures and Algorithms for Information Processing Lecture 5: Trees Copyright © 1999, Carnegie Mellon. All Rights Reserved.

Preorder Traversal 1 Process the root 2 Process the nodes in the left subtree 3 Process the nodes in the right subtree 90 -723: Data Structures and Algorithms for Information Processing Lecture 5: Trees Copyright © 1999, Carnegie Mellon. All Rights Reserved.

Preorder Print public void preorder. Print() { System. out. println(data); if (left != null) left. preorder. Print(); if (right != null) right. preorder. Print(); } 90 -723: Data Structures and Algorithms for Information Processing Lecture 5: Trees Copyright © 1999, Carnegie Mellon. All Rights Reserved.

Inorder Traversal 1 Profess the nodes in the left subtree 2 Process the root 3 Process the nodes in the right subtree 90 -723: Data Structures and Algorithms for Information Processing Lecture 5: Trees Copyright © 1999, Carnegie Mellon. All Rights Reserved.

Inorder Print public void preorder. Print() { if (left != null) left. preorder. Print(); System. out. println(data); if (right != null) right. preorder. Print(); } 90 -723: Data Structures and Algorithms for Information Processing Lecture 5: Trees Copyright © 1999, Carnegie Mellon. All Rights Reserved.