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CS202 Fall 2012 Lecture 17 – 10/30

Trees Prof. Tanya Berger-Wolf

www.cs.uic.edu/~202

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Array - Linked list – Queue - Stack runtime

Operation add to front add to back add at given index clear get find index of an object remove front element remove back element remove at given index set size toString

Array list O(n) O(1) O(n) O(n) O(1) O(n) O(n) O(1) O(n) O(1) O(1) O(n)

Linked list O(1) O(1) O(n) O(1) O(n) O(n) O(1) O(1) O(n) O(n) O(1) O(n)

Queue

O(1)

O(1)

O(1)

O(1) O(n)

Stack

O(1)

O(1)

O(1)

O(1) O(n)

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Application: word search

Write an application that reads in the text of all the web pages in a website (say, www.cs.uic.edu) and then lets the user type words, and tells whether those words are contained in those pages or not.

1.  How would we implement this with a List? 2.  Would this be a good or bad implementation? 3.  Does the ordering of the elements in the List affect the

algorithm? Could we use this information to our advantage?

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List implementation

lists are bad for certain operations 1.  insertion at arbitrary index:

add(index, element)

2.  searching for an element: contains(element), indexOf(element)

3.  removal from arbitrary index: remove(index)

all these operations are O(n) on lists! (bad)

a better data structure to implement this ADT is called a balanced binary search tree;

let's examine trees now…

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Trees

tree: a set of linked nodes; a node may link to more than one other node 1.  an extension / generalization of linked lists

a tree has a starting node called a root ; all other nodes are reachable from the root by the links between them

a node in a tree that does not link to other nodes is called a leaf

Goal: use a tree to build a collection that has O(log n) time for many useful operations

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Tree diagram

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Visualizing trees

every node links to a set of subtrees root of each subtree is a child of root r. r is the parent of each subtree.

r

T1

T2 T3

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Tree terminology

leaf: node with no children siblings: two nodes with the same parent.

path: a sequence of nodes n1, n2, … , nk such that ni is the parent of ni+1 for 1 ≤ i < k 1.  the length of a path is the number of edges in the path, or 1 less

than the number of nodes in it

depth or level: length of the path from root to the current node (depth of root = 0)

height: length of the longest path from root to any leaf

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Tree path length and depth

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Tree example

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CS202 Fall 2012 Lecture – 11/1

Trees

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Trees in computer science

family genealogy

organizational charts 1.  corporate, government, military

folders/files on a computer

AI: decision trees

compilers: parse tree a = (b + c) * d;

d +

* a

=

c b

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Trees in file systems

C:\

hw2

tcss342

hw1 proj1

MyMail

school

D101

pers

one.java calc.java test.java

each folder or file is a node 1.  subfolders = children

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Tree implementation

class TreeNode {

public Object element;

public TreeNode firstChild;

public TreeNode nextSibling;

}

C:\

hw2

tcss342

hw1 proj1

MyMail

school

D101

pers

one.java calc.java test.java

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Tree traversals

traversal: visiting every node in a tree to process every element in it

three common traversal orderings (each one begins at the root):

1.  preorder traversal: the current node is processed, then the node's child subtrees are traversed, in order

2.  in-order traversal: the node's first child's subtree is traversed, then the current node itself is processed, then the node's remaining subtrees are traversed

3.  postorder traversal: the node's child subtrees are traversed in order, and lastly the current node is processed

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Preorder traversal

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Preorder traversal example

procedure preorderTraverse(r) output(r) for each child c of r from left to right, preorderTraverse(c)

Output: a b e j k n o p f c d g l m h i 18

In-order traversal

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In-order traversal example

Output: a b e j k n o p f c d g l m h i

procedure inorderTraverse(r) inorderTraverse(first child of r) output(r) for each child c of r from left to right, excluding first child, inorderTraverse(c)

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Postorder traversal

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Postorder traversal example

procedure postorderTraverse(r)

for each child c of r from left to right,

postorderTraverse(c)

output(r)

Output: j n o p k e f b c l m g h i d a 22

Binary trees

binary tree: a tree where all nodes have at most two children

public class BinaryTree<E> { private TreeNode root;

...

private static class TreeNode { public E element; public TreeNode left; public TreeNode right; } }

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3 2

1

5 4

3

4

2

7

1

5

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Binary tree traversals

three common binary tree traversal orderings (each one begins at the root):

1.  preorder traversal: the current node is processed, then the node's left subtree is traversed, then the node's right subtree is traversed (CURRENT-LEFT-RIGHT)

2.  in-order traversal: the node's left subtree is traversed, then the current node itself is processed, then the node's right subtree is traversed (LEFT-CURRENT-RIGHT)

3.  postorder traversal: the node's left subtree is traversed, then the node's right subtree is traversed, and lastly the current node is processed (LEFT-RIGHT-CURRENT)

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Binary tree preorder traversal order: C F T B R K G

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Binary tree in-order traversal

order: B T R F K C G

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Binary tree postorder traversal order: B R T K F G C

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Infix, prefix, and postfix notation representation of math expressions as a binary tree 1.  operators have their left and right operands as subtrees 2.  literal values are stored as leaves

notations 1.  prefix: Polish notation 2.  infix: standard notation 3.  postfix: reverse Polish notation

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Expression example

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Expression tree example

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Evaluating

Evaluate this postfix expression: 7 2 3 * - 4 ^ 2 5 / +

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Binary search trees

binary search tree (BST): a binary tree where every node n satisfies the following properties: 1.  every element in n's left subtree has a value less than n's

element value 2.  every element in n's right subtree has a value greater than n's

element value 3.  n's left and right subtrees are binary search trees

BSTs' elements are stored in sorted order, which is helpful for searching

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Binary search tree examples

Which of the following two trees are BSTs?

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11 7 1

8 4

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11 4 1

8 3

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Why isn't this a BST?

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11 5

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BST operations

a BST allows us to use a tree to implement a set collection with operations like the following: 1.  contains(element) 2.  add(element) 3.  getHeight 4.  getMin, getMax 5.  removeMin, removeMax 6.  remove(element) 7.  printInOrder, printPreOrder, printPostOrder

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Searching a BST

We must be able to search a binary search tree for a given element's value, to implement the contains operation

Basic idea: compare the element e to be found to the element in the current node of the tree 1.  if they are equal, we are done 2.  if e is smaller, examine the left subtree 3.  if e is greater, examine the right subtree

4.  when can we stop searching?

BST methods are best implemented recursively

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Implementing add

Basic idea: to add element e, find the node n that should be e 's parent, and set n 's child to be a new node containing e

to find parent node n: 1.  if e is smaller, examine the left subtree 2.  if e is greater, examine the right subtree 3.  if we've hit a dead end, we are done; add here

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Implementing add

Traverse from root to expected parent; place a new tree node as parent's left or right child

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Implementing getMin/getMax

To find the maximum element in the BST, we follow right children until we reach null

To find the minimum element in the BST, we follow left children until we reach null

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Implementing remove

Removing from a BST is challenging!

Removing an item disrupts the tree structure 1.  We can't just delete a node like we could with a linked list. 2.  The tree node could have one or two children, so it's not

immediately clear what should become of them.

Basic idea: Find the node that is to be removed. Then "fix" the tree so that it is still a binary search tree.

Three cases: 1.  node to be removed has no children 2.  node to be removed has one child subtree 3.  node to be removed has two child subtrees

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Implementing remove

no children: just set parent's child reference to null one child: replace the removed node with its subtree

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Implementing remove

two children: replace the node with its successor (leftmost element of right subtree), then remove the successor from the tree

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Tree size, height, runtime

for a binary tree t , of size n : 1.  what is the maximum height of t ? n (why?) 2.  what is the minimum height of t ? log n (can you prove it?) 3.  what is the average height of t ? log n

for operations add, contains, remove, getMin, getMax, removeMin, removeMax: 1.  what is their runtime proportional to? (height of the tree) 2.  based on the numbers above, what is the average Big-Oh for

common tree operations? (log n)