Comp 245Data Structures

Trees

Introduction to the Tree ADT

A tree is a non-linear structure. A treenode can point to 0 to N other

nodes. There is one access point to the tree;

it is called the root. A tree is recursive in nature.

Terminology Using a Binary Tree

Root

Child

Parent

Leaf

Height

Level

Full, Complete, Balanced

A FULL Tree A COMPLETE Tree A BALANCED Tree

Traversing a Tree

PRE – order

(v)isit – (l)eft – (r)ight VLR

POST – order

(l)eft – (r)ight – (v)isit LRV

IN – order

(l)eft – (v)isit – (r)ight LVR

A PRE-order Traversal(VLR)

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This traversal is considered “top-down”

A POST-order Traversal(LRV)

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This traversal is considered “bottom-up”

An IN-order Traversal(LVR)

This traversal is considered “left to right”

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Traversal Practice

Traversal Practice

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Post478529631

In427581396

Sample Code of a Traversal

void Tree::InOrder(TreePtr P){

if (P != NULL){

InOrder(P->Left);Process(P);InOrder(P->Right);

}}

Implementing a Binary TreeLinked

struct Node;

typedef SomeDataType TreeType;

typedef Node* TreePtr;

struct Node{

TreeType info;TreePtr Left, Right;

};

Defining a Binary TreeLinked

class Tree{ public:

Tree();~Tree();bool Empty();bool Insert(TreeType);bool Delete(TreeType);void Traverse();

private:void InOrder(TreePtr);TreePtr Root;

};

Binary Search TreesBST

A special type of tree that is very useful!

Main characteristic:Given any node P, the left child is lesser than or

equal to P; the right child is greater than P.

The efficiency of a BST ranges from logarithmic time to linear time.

Example of BST Efficiency

How many accesses to find R?

How many accesses to find R?

BST Efficiency

Assuming a tree is balanced, it’s efficiency is approximately log2N where N is the number of elements in the tree.

Example:There are 1000 elements in a BST, it’s efficiency therefore

is approximately log21000 = 9.9 or 10. This means that it will take in the absolute worst case, 10 accesses to find a value in the tree. If you contrast this to an ordered list, it will take 1000 accesses in the worst case and 500 in the average case to find an element!!

If a tree is not balanced, it’s efficiency will degenerate!

BST Operation - Insertion

The Insert function can be highly efficient.

The new value is always inserted as a leaf node!

BST Operation – InsertionPractice: Build a BST

Build a BST from these values:

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Inserting a Node into a BST

Create a node (Test for success) Store data, set right and left pointers

null (it will be a leaf) Search tree for insertion point, keep

track of node which will become the parent.

Attach this node to parent. Return success or failure of operation.

Deleting a Node from a BST

There are three cases to account for:LeafOne ChildTwo Child

The algorithm requires a Search to find the node to delete, determining the specific case, and then executing the deletion.

Leaf Case

How do you know the node is a leaf?

This routine will require 1) a pointer to the node to be deleted and 2) a pointer to the parent.

Delete Leaf Code

void Bst::DeleteLeaf (TreePtr P, TreePtr Parent){

//check for rootif (Parent == NULL)

Root = NULL;else

if (Parent->Left == P)Parent->Left = NULL;

elseParent->Right = NULL;

delete P;}

One Child Case

How do you know the node has one child?

This routine will require 1) a pointer to the node to be deleted and 2) a pointer to the parent.

Delete One Child Code

void Bst::DeleteOneChild (TreePtr P, TreePtr Parent)

{

1) save pointer to subtree, must be re-attached

2) check for root case

3) re-attach subtree to parent

4) delete P

}

Two Child Case

How do you know the node has two children?

This routine will require only a pointer to the node to be deleted.

Finding the Closest Predecessor

From the two child node to be deleted, take one step left and go as far right as possible. This node is the closest predecessor.

Place this value in the node to be deleted.

The closest predecessor will be deleted by calling DeleteLeaf or DeleteOneChild.

Delete Two Children Case

void Bst::DeleteTwoChild (TreePtr P)

{

1) Find closest predecessor (cp), keep track of

parent to cp!!

2) Copy cp->info to P->info

3) Call DeleteLeaf or DeleteOneChild for cp

}

Traversal UsagePreorder

The preorder traversal can be used to effectively save a tree to file that can be reconstructed identically. This type of traversal can be used to copy a tree also.

MikeDonHarryGregTimPaulWayne

Traversal UsageInorder

The inorder traversal can be used to obtain a sorted

list from a BST.

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Traversal UsagePostorder

The postorder traversal can be used to delete a tree. A tree needs to be deleted from the bottom up because every node at the point of deletion is a leaf.

Order of DeletionGregHarryDonPaulWayneTimMike

Binary Tree ImplementationArray Based – method 1

The first method will store information in the tree traveling down levels going left to right.

Given this storage technique, a node stored at slot I in the array will have it’s left child at 2I + 1, and the right child will be at 2I + 2.

A parent can be found at (I – 1)/2.

Binary Tree ImplementationArray Based – method 2

The second method will have an array of structs. Each struct will contain the information and left and right pointer fields. The pointer fields will simply be index values within the array.

Each new value is added at the end of the array as a leaf and the pointer to it’s parent adjusted.

N-Ary TreesFirst Child-Sibling Implementation