Data Structures - CSCI 102 Binary Tree In binary trees, each Node can point to two other Nodes and...

Data Structures - CSCI 102

Binary TreeIn binary trees, each Node can point to two other Nodesand looks something like this:

template <typename T>class BTNode {

public://The data section

BTNode

T data;left right

//The linksBTNode<T>* left;BTNode<T>* right;

Binary Tree

Trees grown downward in computer science

Binary Tree Terminology

Each BTNode has 0, 1 or 2 Child Nodes that are below it inthe tree

A BTNode that has a child is called that child’s Parent Node

Parent nodes point to child nodes via a branch (child nodesdon’t need to point to parents, but they can if they want)

Binary TreesA is the Parent of B and C

B is the Left Child of A

C is the Right Child of A

Branch

The Root Node is the only BTNode in the tree that has noparent node

Binary Tree Terminology

A BTNode is a Leaf Node if it has no children

The Level of a BTNode in the tree is the number ofbranches from the root to the node

The Height of the tree is the number of nodes on thelongest path from root to leaf

Binary Trees

A is the Root Node

C, D, E are Leaf Nodes

Level(A) = 0

Level(B) = 1Level(D) = 2

The Height of the tree is 3

What kind of operations do we need to perform on a binarytree?

Binary Trees

height()size()search()insert()delete()print()

NOTE: Pretty much every operation requires a way totraverse the tree

Binary Tree HeightIf you had to calculate the height of a binary tree, howwould you do it?

Binary Tree HeightHeight = 1 + max(Height(Root->left),Height(Root->right))

Height ofRoot->left

Root->right

Height of

Binary Tree TraversalIf you had to print out all the nodes in a binary tree, howwould you do it?

If you had to print out all the nodes in a binary tree, howwould you do it?

Binary Tree Traversal

At each node in the tree, you have three choices:

These choices leads to different kinds of traversals

Print the current nodeGo to the left childGo to the right child

In Order: print left, print current, print rightPre Order: print current, print left, print rightPost Order: print left, print right, print current

Binary Tree TraversalIn Order Traversal:

Traverse A’s left subtreePrint ATraverse A’s right subtree

Order of output:GDBHEIACF

Pre Order Traversal:Print ATraverse A’s left subtreeTraverse A’s right subtree

Order of output:ABDGEHICF

Post Order Traversal:Traverse A’s left subtreeTraverse A’s right subtreePrint A

Order of output:GDHIEBFCA

Binary Tree SearchIf you had to find a particular value in a binary tree, howwould you do it?

Search still requires us to potentially look at everysingle node in the treeStill O(n)

So far binary trees are interesting, but they don’t provideany extra benefit

Binary Tree Issues

If you’re going to insert a new Node, where should youput it in the tree?

What’s missing here is some extra organization that makesit easier to find things

template <typename T, typename U>class BTNode {

public:T key; //The key dataU data; //The data

left right//The linksBTNode<T,U>* left;BTNode<T,U>* right;

Binary TreeBTNode can also have a separate key and data

key is used to order the nodes in the treedata actually holds the stuff we care about

BTNode

To speed up insertion, removal and search, modify the ideaof a Binary Tree to create a Binary Search Tree (BST)

Binary Search Trees

Binary Search Trees have one major organizationalproperty (the binary search tree property):

Let X be a Node in a binary search tree

Every Node X is comparable to every other Node via afield we call the key (for now, "data" is the key)If Y is a Node in the left subtree of X:key[Y] <= key[X]

key[Y] > key[X]

If keys must be unique, then we have key[Y] < key[X]You can implement your BST to enforce thisWe will assume that this is the case here

If Y is a Node in the right subtree of X:

Binary Search Tree Search

If you had to find a particular value in abinary search tree, how would you do it?

Ex: find 58

If you had to find a particular value in abinary search tree, how would you do it?

Ex: find 58Start at root

58 == 60? No58 < 60? Yes, go left

Current

RootIf you had to find a particular value in abinary search tree, how would you do it?

58 == 60? No58 < 60? Yes, go left

Current

58 == 50? No58 < 50? No58 > 50? Yes, go right

RootIf you had to find a particular value in abinary search tree, how would you do it?

58 == 60? No58 < 60? Yes, go left58 == 50? No58 < 50? No

Current

58 > 50? Yes, go right58 == 58? Yes

Start at the rootWhat will our search algorithm be?

If the key we want is less than the key of the currentnode, search the left subtree

Is the current node what we want?

Otherwise, search the right subtree

If yes, we’re doneIf no...

Should our solution be recursive or iterative?

Binary Search Tree Min/MaxHow would you find the minimum/maximum values in abinary search tree?

Start at the rootWhat will our findMin algorithm be?

Binary Search Tree Min/Max

Does the current Node have a left child?If yes, go to the left childIf no, we’re at the minimum

Start at the rootWhat will our findMax algorithm be?

Does the current Node have a right child?If yes, go to the right childIf no, we’re at the maximum

Order of output:30 46 50 58 60 70 77 80

discussed?In Order Traversal

Visits all the nodesin the tree in order

Binary Search Tree TraversalNow that we have an implicit ordering to our tree, whathappens with the traversal algorithms we previously

Binary Search Tree Traversal

Traverse the binary search tree with an in-ordertraversal

Tree SortGiven an unordered list of elements, add them all to abinary search tree

All the nodes come back out sorted!

Binary Search Tree Insertion

New nodes are always added in the place of an existingNULL...they never reposition other nodes

1) Create the new node with its data and set both of itspointers to NULL2) If the tree is empty, make the new node the root3) Walk down the tree node by node like you were doinga search

If new value is less than current node, move to leftsubtree, otherwise move to right subtreePotentially reject duplicates

Binary Search Tree InsertionHow do you insert a new value into a binary search tree?

RootEx: insert 59

Ex: insert 59Make new node

59New Node

Current

Ex: insert 59Make new nodeStart at root59 < 60; go left

59 > 50; go right

59New Node

Current

59New Node

Current

59 > 50; go right59 > 58; go right

59 > 50; go right59 > 58; go rightRight is NULL

Insert new Node

59 > 50; go right59 > 58; go rightRight is NULL

Insert new Node

Binary Search Tree Removal

Binary Search Tree Remove

Search the tree for the node that you want to removeWhen node is found, removal breaks down into 3 distinctcases

1) Remove a node with no children2) Remove a node with one child3) Remove a node with two children

Find the node and its parentRemove a node with no children (case #1)

Set the parent pointer to the node to NULLDelete the node

Case #1 Ex: remove 46

Binary Search Tree RemoveHow do you remove a value from a binary search tree?

To remove

Case #1 Ex: remove 46Search for the value

To remove

Case #1 Ex: remove 46Search for the valueSet parent’s pointer to NULL

To remove

Case #1 Ex: remove 46Search for the valueSet parent’s pointer to NULLDelete the child

RootCase #1 Ex: remove 46Search for the valueSet parent’s pointer to NULLDelete the child

Find the node and its parentRemove a node with one child (case #2)

Set the parent pointer to the node’s only childDelete the node

Case #2 Ex: remove 30

To remove

Case #2 Ex: remove 30Search for the value

To remove

Case #2 Ex: remove 30Search for the valuePointer value’s parent tovalue’s only child

To remove

Delete the node

Find the nodeRemove a node with two children (case #3)

Remove node’s successor or predecessorReplace node’s data with successor/predecessor data

The node in the tree with the biggest value less than xPredecessor(x)

The node in the tree with the smallest value greaterthan x

Successor(x)

RootCase #3 Ex: remove 50

Search for the valueTo remove

Search for the valueFind predecessor

Rightmost value in

To remove

left subtree of node

Predecessor

To remove

Search for the valueFind predecessor

Rightmost value inleft subtree of node

Swap values of nodeand its predecessor

Case #3 Ex: remove 50Search for the valueFind predecessor

Rightmost value in

To remove

left subtree of nodeSwap values of nodeand its predecessor

Delete the node viacase #1 or case #2

Case #3 Ex: remove 50Search for the valueFind predecessor

Rightmost value in

left subtree of nodeSwap values of nodeand its predecessor

Delete the node viacase #1 or case #2

60, 46, 77, 45, 50, 63, 91Insert the following values in an empty tree (in this order)

Binary Search Tree Construction

What does the resulting tree look like?

Binary Search Tree ConstructionInsert the following values in an empty tree (in this order)

60, 46, 77, 45, 50, 63, 91

50 6345

What does the resulting tree look like?Tree is perfectly balanced!

Height = log(n+1)

91, 77, 63, 60, 50, 46, 45

Insert the following values in an empty tree (same values aslast time, but different order)

Binary Search Tree Construction

91Root

Binary Search Tree ConstructionInsert the following values in an empty tree (same values as

last time, but different order)91, 77, 63, 60, 50, 46, 45

What does the resulting tree look like?Insert 91

91Root

Insert 91Insert 77

91Root

63Insert 77

Insert 63

91Root

63Insert 77Insert 63 60Insert 60

91Root

What does the resulting tree look like?Tree is completely unbalanced

63Height = n

Data Structures - CSCI 102 Binary Tree In binary trees, each Node can point to two other Nodes and...

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