Chapter 7. Trees & Binary Trees

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Chapter 7. Trees & Binary Trees

- Set ADT

- Introduction to trees and binary trees

Lecture 14

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Pictures of trees

File System

Pictures of trees

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Pictures of trees

Pictures of treesunrooted

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unrooted Pictures of trees

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Pictures of trees

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Pictures of trees

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Pictures of trees

Pictures of trees

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General Trees

• Nonlinear data structure• Natural way to organise data

• File system• GUI• Databases• Websites• Inheritance relation in Java classes• Books (chapters, sections, subsections, subsubsections)

• “nonlinear” organisational structure• Not just “before” “after”• “above” “below” “part of”

• Relationships are typically• Parent• Children• Ancestors• Descendents• Siblings

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General TreesFormal definition

A tree T is either empty or consists of a node r, the root of T, and a (possibly empty) set of treeswhose roots are the children of r

An edge in T is a pair of nodes (u,v) where u is parent of v or v is parent of u

A tree T is a set of nodes with a parent-child relationshipEach node has one parent, apart from the root (which has no parent)

• a node• an edge• a path• a child• a parent• an ancestor• a descendant• siblings• depth of a node• height of a node• height of a tree• a leaf• an internal node• the root• a subtree

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General Trees

• n nodes• n-1 edges• no cycles• unique path from a node to root

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General Trees

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How might we implement a general tree?

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General Trees

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Therefore we might implement a tree as follows

Node:•<E> element•Node<E> parent•ArrayList<Node<E>> children

Tree<E>•Node<E> root

Note: there could be order amongst the children

General Trees

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Depth

Depth of a node is how far it is from the root.

depth(Node<E> node) if (node.isRoot()) return 0; return 1 + depth(node.parent());

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Height

Height of a node is•If node is a leaf then 0•Otherwise 1 + the maximum of the height of its children

height(Node<E> node) if (node.isLeaf()) return 0; int h = 0; for (Node<E> child : node.children()) h = Math.max(h,height(child)); return 1 + h;

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Preorder Traversal

Visit the parent then visit its children

General Trees

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Preorder Traversal


preorder(Node<E> node) if (node != null) { visit(node); for (Node<E> child : node.children()) preorder(child); }

Whatever “visit” means

General Trees

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Preorder Traversal




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General Trees

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Preorder Traversal




O(n)

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General Trees

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Parenthetic Representation

(a (c (b ((R),(X),(n))),(d)),(p),(Z),(i (g ((f),(h))),(j)))

Also called Caley Notation (I think) and is a preorder print

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Postorder Traversal

Visit the children then visit the parent

Typical use is we want to “assemble” parts

An actual use: du in unix

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Postorder Traversal


postorder(Node<E> node) if (node != null) { for (Node<E> child : node.children()) postorder(child); visit(node); }

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Postorder Traversal


prostorder(Node<E> node) if (node != null) { for (Node<E> child : node.children()) postorder(child); visit(node); }

O(n)

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Postorder Traversal


prostorder(Node<E> node) if (node != null) { for (Node<E> child : node.children()) postorder(child); visit(node); }

O(n)

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General Trees

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Other Traversals

Visit depth 0 nodesVisit depth 1 nodes …Visit depth height nodes

Also Known As (aka) Breadth First Search (bfs)

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Other Traversals

Visit depth 0 nodesVisit depth 1 nodes …Visit depth height nodes

Also Known As (aka) Breadth First Search (bfs)

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Note: preorder = dfs!

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Binary Trees

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Binary TreesA binary tree is a tree in which each node has

• A reference to a left node• A value• A reference to a right node

A binary tree is either empty or Contains a single node r (the root)whose left and right subtrees are binary trees The tree is accessed via a reference to the root.The nodes with both child references null are the leaves.

recursive definition!

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• nodes 0• intenal nodes 0• leaf nodes 0• edges 0• height ?

• nodes 1• intenal nodes 0• leaf nodes 1• edges 0• height 0

• nodes n• intenal nodes (nI) h ≤ nI ≤ 2h-1• leaf nodes (nE) 1 ≤ nE ≤ 2h • edges (e) e = n-1• height (h) h+1 ≤ n ≤2h+1-1

height (h) log2(n+1)-1 ≤ h ≤ n-1


height (h) log2(n+1)-1 ≤ h ≤ n-1

And this is crucial

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A Forest!

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This is also a binary tree

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This tree is quite well balanced.An extreme unbalanced tree might have no right pointers

- would look more like a linked list.Generally tree-based algorithms work most efficientlyon balanced trees.A binary tree in which no value occurs in more than one node is injective. We deal only with injective binary trees.

Balance

A typical binary tree:

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Node of binary treeBinary tree node:

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public class BTNode<E>{ private E element; private BTNode<E> left; private BTNode<E> right;

/**Creates a node with null references*/ public BTNode(){ this(null,null,null); }

/** Creates node with the given element, L and R nodes*/ public BTNode(E e, BTNode<E> l,BTNode<E> r){ element = e; left=l; right=r; }

A question:

How would we represent the nodes of a tree (not binary)?

We don’t know how many children each node has

plus getters and setters

For a binary tree of strings


And what is that?

Definitions

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p - a node of a binary treep.left - node of a binary tree - the left subtreep.right - node of a binary tree - the right subtreeif p.left = q - p is the parent of q and q is the left child of

pif p.right = q - p is the parent of q and q is the right child

of p.

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Left subtree of p

Right subtree of p

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Traversals

A traversal of a binary tree is a sequence of nodes of the tree.

Special traversals:

1. Inorder traversal - defined recursively

The inorder traversal of an empty tree is the empty sequenceThe inorder traversal of a non-empty tree is: the inorder traversal of the left subtree (a sequence) the root value (a sequence with one member) the inorder traversal of the right subtree (a sequence)

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Example (inorder traversal)

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Inorder traversal is a b c d e f g h i j

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Traversals contd.2. Preorder traversal – defined recursively

The preorder traversal of an empty tree is the empty sequenceThe preorder traversal of a non-empty tree is:

the root value (a sequence with one member)the preorder traversal of the left subtree (a sequence)the preorder traversal of the right subtree (a sequence)

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Preorder traversal is e c b a d i g f h j

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Traversals contd.

Postorder traversal defined recursively:

The postorder traversal of an empty tree is the empty sequenceThe postorder traversal of a non-empty tree is:

the postorder traversal of the left subtree (a sequence)the postorder traversal of the right subtree (a sequence)the root value (a sequence with one member)

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Postorder traversal is a b d c f h g j i e

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An Euler Walk

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An ArrayList Representation of a Binary Tree

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Nodes have an integer positionWe put the nodes in an Array or ArrayList

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We have an ArrayList<Node<E>> T A Node<E> has an integer position attributeThe root has position 1, i.e. T[1] is the root nodeT[i].position() == i

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We have an ArrayList<Node<E>> T A Node<E> has an integer position attributeThe root has position 1, i.e. T[1] is the root nodeT[i].position() == i

Left of T[i] is T[i*2]Right of T[i] is T[i*2 + 1]Parent of T[i] is T[i/2]

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An Expression Tree (or species tree)Putting an expression into a tree

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We use two stacks•S1 of Node<E> •S2 of String

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We use two stacks•S1 of Node<E> •S2 of String

if we read “(“ S1.push(new Node(null,”!”,null))if we read is in {+,-,*,/} S2.push(operator)if we read “)” •Node r = S1.pop()•Node l = S1.pop()•Node node = S1.pop()•String op = S1.pop()•node.setLeft(l); node.setElement(op); node.setRight(r)•S1.push(node)if we read String s and it is something else (a leaf) S1.push(new Node(null,s,null))

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No Expense Spared

Chapter 7. Trees & Binary Trees

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Transcript of Chapter 7. Trees & Binary Trees