Chapter 7. Trees & Binary Trees

Lecture 14 Chapter 7. Trees & Binary Trees Set ADT Introduction to trees and binary trees ADS2 Lecture 14

Pictures of trees File System

Pictures of trees

ADS2 Lecture 14

Pictures of trees unrooted

unrooted Pictures of trees ADS2 Lecture 14 12

Pictures of trees ADS2 Lecture 14 13

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 ADS2 Lecture 14

Formal definition General Trees A tree T is a set of nodes with a parent-child relationship Each node has one parent, apart from the root (which has no parent) A tree T is either empty or consists of a node r, the root of T, and a (possibly empty) set of trees whose 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 ADS2 Lecture 14

a e c i b d g j f h R General Trees • 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 Z p n X

a e c i b d g j f h R General Trees • n nodes • n-1 edges • no cycles • unique path from a node to root Z p n X

How might we implement a general tree? ADS2 Lecture 14

a e c i b d g j f h R General Trees Z p Therefore we might implement a tree as follows • Tree<E> • Node<E> root n X • Node: • <E> element • Node<E> parent • ArrayList<Node<E>> children Note: there could be order amongst the children ADS2 Lecture 14

a e c i b d g j f h R Depth General Trees • Tree<E> • Node<E> root • Node: • <E> element • Node<E> parent • ArrayList<Node<E>> children Z p Depth of a node is how far it is from the root. n X depth(Node<E> node) if (node.isRoot()) return 0; return 1 + depth(node.parent());

a e c i b d g j f h R Height General Trees • Tree<E> • Node<E> root • Node: • <E> element • Node<E> parent • ArrayList<Node<E>> children Z p • Height of a node is • If node is a leaf then 0 • Otherwise 1 + the maximum of the height of its children n X 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;

a e c i b d g j f h R Preorder Traversal General Trees • Tree<E> • Node<E> root • Node: • <E> element • Node<E> parent • ArrayList<Node<E>> children Z p n X Visit the parent then visit its children

a e c i b d g j f h R Preorder Traversal General Trees • Tree<E> • Node<E> root • Node: • <E> element • Node<E> parent • ArrayList<Node<E>> children Z p Visit the parent then visit its children n X preorder(Node<E> node) if (node != null) { visit(node); for (Node<E> child : node.children()) preorder(child); } Whatever “visit” means

a e c i b d g j f h R Preorder Traversal General Trees • Tree<E> • Node<E> root • Node: • <E> element • Node<E> parent • ArrayList<Node<E>> children Z p a c b R X n d p Z i g f h j Visit the parent then visit its children n X preorder(Node<E> node) if (node != null) { visit(node); for (Node<E> child : node.children()) preorder(child); } Whatever “visit” means

a e c i b d g j f h R Preorder Traversal General Trees • Tree<E> • Node<E> root • Node: • <E> element • Node<E> parent • ArrayList<Node<E>> children Z p a c b R X n d p Z i g f h j Visit the parent then visit its children n X preorder(Node<E> node) if (node != null) { visit(node); for (Node<E> child : node.children()) preorder(child); } O(n) Whatever “visit” means

ADS2 Lecture 14

a e c i b d g j f h R Parenthetic Representation General Trees • Tree<E> • Node<E> root • Node: • <E> element • Node<E> parent • ArrayList<Node<E>> children Z p n X (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

Chapter 7. Trees & Binary Trees