Trees A non-linear implementation for collection classes

Linear structures Arrays and linked lists are linear: Obvious first and last elements Obvious order for traversal

Non-linear structures Other organizations are possible, (e.g., file organization on disk is a tree) May be more efficient No obvious first and last elements No obvious order for traversal c: \drivers \documents and settings \program files \adobe \canon \google

Tree as graph Connected graph with n nodes and n-1 edges -> no cycles

Rooted tree – any root -directed edges -one node is root -all nodes accessible from root -root has indegree 0 -leaf nodes have outdegree 0 -branch nodes have in and out edges

Relative terminology Relative to a node Parent node Ancestors Child node Descendents Sibling node

Measuring location by path length Depth (from root) 2 Height (above deepest descendent) 2

Recursive definition of list Empty list (0 nodes) OR Head node and 0 or 1 non-empty sublist

Recursive definition of list Empty list (0 nodes) OR Head node and 0 or 1 non-empty sublist

Recursive definition of tree Empty tree (0 nodes) OR Root node and 0 or more non-empty subtrees (child node and its descendents)

Recursive definition of tree

Tree application – file system Directory structure is tree rooted at / (Windows explorer)

A Unix directory Data Structures & Problem Solving using JAVA/2E Mark Allen Weiss © 2002 Addison Wesley

The Unix directory with file counts Data Structures & Problem Solving using JAVA/2E Mark Allen Weiss © 2002 Addison Wesley Task: count total files (in parentheses at each node)

The Unix directory with file counts Data Structures & Problem Solving using JAVA/2E Mark Allen Weiss © 2002 Addison Wesley Task: count total files (in parentheses at each node) recursively

Number of files in subtrees Data Structures & Problem Solving using JAVA/2E Mark Allen Weiss © 2002 Addison Wesley

Number of files in subtrees Data Structures & Problem Solving using JAVA/2E Mark Allen Weiss © 2002 Addison Wesley

Traversal: how to organize activity on all nodes in a collection? List – process sequentially by 1.iteration 2.recursion

Tree – no obvious order to process ? Traversal: how to organize activity on all nodes in a collection?

Traversal of trees Problem: how to visit all nodes; what order to visit: Many traversals for different purposes: Breadth first Depth first Preorder, postorder (recursive) …more

Preorder Traversal Recursive definition: Visit root before children 1.Visit root 2.Visit subtrees (left to right)

Preorder Traversal Recursive definition: Visit root before children 1.Visit root 2.Visit subtrees (left to right)

Recursive definition of tree

Postorder Traversal Recursive definition: Visit root after children 1.Visit subtrees (left to right) 2. Visit root

Postorder Traversal Recursive definition: Visit root after children 1.Visit subtrees (left to right) 2. Visit root

Traversal examples W RS A TFC OX D I UP B E

Implementing trees List –Node – data & reference Tree –Node – data & references class ListNode {int data; ListNode next; } class TreeNode {int data; TreeNode child1, child2, child3; }

Implementing trees Problem of representing a node: How many edges to allow for? e.g., Class TreeNode { int data; TreeNode child1, child2, child3; }

Implementing trees Problem of representing a node: Three solutions: 1.allow lots of child edge references 2.restrict tree structure to two child edges 3.use first child/next sibling representation

First child/next sibling Class TreeNode { int data; TreeNode child, sibling; } Problem: Makes path to most nodes longer

Binary Trees result – for many purposes, two references are sufficient  binary trees with -left child -right child class BNode { int data; BNode left, right; }