Main Index Contents 11 Main Index Contents Tree StructuresTree Structures (3 slides) Tree Structures Tree Node Level and Path Len. Tree Node Level and

1 Main IndexMain Index ContentsContents1 Main IndexMain Index ContentsContents

Tree Structures (3 slides) (3 slides)

Tree Node Level and Path Len.

(5 (5 slides)slides)

Binary Tree Definition

Selected Samples / Binary Trees

Binary Tree Nodes

Binary Search Trees

Locating Data in a Tree

Removing a Binary Tree Node

stree ADT (4 slides)

Using Binary Search Trees

- Removing Duplicates

Update Operations (3 slides) (3 slides)

Removing an Item From a Binary Tree

(7 slides) (7 slides)

Chapter 10 Chapter 10 – – Binary TreesBinary Trees

Summary Slides (5 slides)


Tree StructuresTree Structures

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10 Main IndexMain Index ContentsContents

Binary Tree DefinitionBinary Tree Definition A binary tree T is a finite set of nodes with one

of the following properties:– (a) T is a tree if the set of nodes is empty.

(An empty tree is a tree.)– (b) The set consists of a root, R, and exactly two

distinct binary trees, the left subtree TL and the right subtreeTR. The nodes in T consist of node R and all the nodes in TL and TR.


Selected Samples of Binary Selected Samples of Binary TreesTrees

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Binary Search TreesBinary Search Trees

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Current Node Action --LOCATING DATA IN A TREE-Root = 50 Compare item = 37 and 50

37 < 50, move to the left subtreeNode = 30 Compare item = 37 and 30

37 > 30, move to the right subtreeNode = 35 Compare item = 37 and 35

37 > 35, move to the right subtreeNode = 37 Compare item = 37 and 37. Item found.


Removing a Binary Search Tree Removing a Binary Search Tree NodeNode

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CLASS stree Constructors “d_stree.h”

stree();Create an empty search tree.

stree(T *first, T *last);Create a search tree with the elements from the

pointer range [first, last).

CLASS stree Opertions “d_stree.h”

void displayTree(int maxCharacters);Display the search tree. The maximum number of

characters needed to output a node value is maxCharacters.bool empty();

Return true if the tree is empty and false otherwise.



int erase(const T& item);Search the tree and remove item, if it is present;

otherwise, take no action. Return the number of elements removed.

Postcondition: If item is located in the tree, the size of the tree decreases by 1.

void erase(iterator pos);Erase the item pointed to the iterator pos.

Precondition: The tree is not empty and pos points to an item in the tree. If the iterator is invalid, thefunction throws the referenceError exception.

Postcondition: The tree size decreases by 1.



void erase(iterator first, iterator last);Remove all items in the iterator range [first, last).

Precondition: The tree is not empty. If empty, the function throws the underflowError exception.

Postcondition: The size of the tree decreases by the number of items in the range.

iterator find(const T& item);Search the tree by comparing item with the data

values in a path of nodes from the root of the tree. If a match occurs, return an iterator pointing to the matching value in the tree. If item is not in the tree, return the iterator value end().



Piar<iterator, bool> insert(const T& item);If item is not in the tree, insert it and return an iterator-bool pair where the iterator is the location of the

newelement and the Boolean value is true. If item is already in the tree, return the pair where the iterator locates the existing item and the Boolean value is false.

Postcondition: The size of the tree is increased by 1 if item is not present in the tree.

int size();Return the number of elements in the tree.


Using Binary Search Trees Using Binary Search Trees Application: Removing DuplicatesApplication: Removing Duplicates

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Update Operations: Update Operations: 11stst of 3 steps of 3 steps

1)- The function begins at the root node and compares item 32 with the root value 25. Since 32 > 25, we traverse the right subtree and look at node 35.

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Update Operations: Update Operations: 22ndnd of 3 steps of 3 steps

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Summary Slide 1Summary Slide 1

§- trees

- hierarchical structures that place elements in nodes along branches that originate from a root.

- Nodes in a tree are subdivided into levels in which the topmost level holds the root node.

§- Any node in a tree may have multiple successors at the next level. Hence a tree is a non-linear structure.

- Tree terminology with which you should be familiar:

parent | child | descendents | leaf node | interior node | subtree.



§- Binary Trees- Most effective as a storage structure if it has high

density §- ie: data are located on relatively short paths from the

root.

§- A complete binary tree has the highest possible density

- an n-node complete binary tree has depth int(log2n).

- At the other extreme, a degenerate binary tree is equivalent to a linked list and exhibits O(n) access times.



§- Traversing Through a Tree- There are six simple recursive algorithms for tree

traversal.

- The most commonly used ones are:

1) inorder (LNR)

2)postorder (LRN)

3)preorder (NLR).

- Another technique is to move left to right from level to level.§- This algorithm is iterative, and its implementation

involves using a queue.



§- A binary search tree stores data by value instead of position

- It is an example of an associative container.§- The simple rules

“== return”

“< go left”

“> go right”

until finding a NULL subtree make it easy to build a binary search tree that does not allow duplicate values.



§- The insertion algorithm can be used to define the path to locate a data value in the tree.

§- The removal of an item from a binary search tree is more difficult and involves finding a replacement node among the remaining values.

Documents

Main Index Contents 11 Main Index Contents Tree StructuresTree Structures (3 slides) Tree Structures Tree Node Level and Path Len. Tree Node Level and