Lecture 10,11

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FUIEMS Malik Imran Daud

Data Structures

Trees

Lecture 10,11

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What is a tree?What is a tree?

In computer science, a tree is an abstract model of

a hierarchical structure

A tree consists ofnodes with aparent-child

relation

Previous data structures placed data in linear

order.

Some data organizations require categorizing datainto groups, subgroups

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Tree TerminologyTree Terminology Trees are a hierarchical data structure ofTrees are a hierarchical data structure ofnodesnodes

Each position in the tree is called aEach position in the tree is called a nodenode

Nodes are linked byNodes are linked by edgesedges

4

2 5

1 3

edge

nodes

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Applications

Organization charts

File systems

Programming environments A class hierarchy in programming languages that support singleA class hierarchy in programming languages that support single

inheritance (e.g. Java)inheritance (e.g. Java)

Document Object Model (for HTML and XMLDocument Object Model (for HTML and XML

Artificial Intelligence (Decision making etc)

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tree is hierarchical classification

A tree is hierarchical classification.

Data items appear at various levels within the organization

E.g., directory structure:

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tree is hierarchical classification

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Decision TreesDecision Trees

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Decision tree based upon expertDecision tree based upon expert

systemsystem

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Tree Terminology (Parent /Tree Terminology (Parent /

Child)Child) Root: node without parent

AAparent nodeparent node references one or morereferences one or more

nodesnodes (childrennodes(childrennodes) that are lower) that are lower

in the tree hierarchyin the tree hierarchy

Ifnode u is the parent ofnode v, then vIfnode u is the parent ofnode v, then v

is the child of uis the child of u

Except for the root (no parent), everyExcept for the root (no parent), every

node has exactly one parent (bynode has exactly one parent (by

definition)definition)

A tree has only one root nodeA tree has only one root node

u

v

root

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Tree Terminology (InternalTree Terminology (Internal

node)node) Anode is internal if it has one or more childrenAnode is internal if it has one or more children

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Tree Terminology (Leaf /ExternalTree Terminology (Leaf /External

node)node)

Anode is a leaf if it has no childrenAnode is a leaf if it has no children

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Tree Terminology (Ancestor /Tree Terminology (Ancestor /

Descendent)Descendent)

An ancestor of a node is either the nodes parent or anyAn ancestor of a node is either the nodes parent or anyancestor of the nodes parent (this is recursive)ancestor of the nodes parent (this is recursive)

The root is an ancestor of each othernodeThe root is an ancestor of each othernode

Descendant of a node: child, grandchild, grand-grandchild

u

v

u

v

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Tree Terminology (Subtree)Tree Terminology (Subtree)

Anode plus the collection ofnodes beneath it is called aAnode plus the collection ofnodes beneath it is called asubtreesubtree

A tree may be divided intoA tree may be divided into subtreessubtrees..

Hence, a subtree is composed of aHence, a subtree is composed of a node and all of that nodesnode and all of that nodesdescendants.descendants.

Subtrees themselves can be further divided into other subtrees.Subtrees themselves can be further divided into other subtrees.

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Tree TerminologyTree Terminology

(Subtree)(Subtree) The subtree of T rooted at node v is the tree consisting of all theThe subtree of T rooted at node v is the tree consisting of all thedescendents of v in T (including v)descendents of v in T (including v)

tree consisting of a node and its descendants

v

v

Root of subtree

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Tree Terminology (Depth of aTree Terminology (Depth of a

node)node)

The depth of a node v in T is the number of ancestorsThe depth of a node v in T is the number of ancestors

of v, excluding v itself.of v, excluding v itself.

More formally:More formally:

If v is the root, the depth of v is 0If v is the root, the depth of v is 0

depth ofv = 1 depth ofv = 3

v

v

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Tree Terminology( HeightTree Terminology( Height

/depth of a tree)/depth of a tree) The number of edges in the longest path from the root to a leafThe number of edges in the longest path from the root to a leafis theis the depthdepth (or(orheightheight) of the tree) of the tree

maximum levels of a tree

tree depth = 3

tree depth = 2

tree depth = 0

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Types of TreesTypes of Trees

General treeGeneral tree a node cana node can

have any number of childrenhave any number of children

Binary treeBinary tree a node cana node can

have at most two childrenhave at most two children

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Binary TreeBinary Tree

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Binary TreesBinary Trees The simplest form ofThe simplest form oftreetree is a Binary Tree in which eachis a Binary Tree in which each

node has at most 2 child nodes.node has at most 2 child nodes.

ABinary Tree consists ofABinary Tree consists of

(a) A(a) Anodenode (called the(called the rootroot node) andnode) and

(b)(b) LeftLeft andand right subtreesright subtrees

Both the subtrees are themselves binary treesBoth the subtrees are themselves binary trees Note: this is a recursive definitionNote: this is a recursive definition

(Anode cant have more than 2 children)(Anode cant have more than 2 children)

General treeBinary tree

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A binary tree is either empty or has the following

form:

Where Tleft and Tright are binary trees.

root

TLTR

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Binary TreesBinary Trees

Full binary tree: All leaves on the same level and

every node has either zero or two children.

Complete binary tree: Leaves are filled from left to

right on one level before moving to next level.

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Full binary tree of height (depth)Full binary tree of height (depth) hh: all nodes at a: all nodes at a

height less thanheight less thanhh have exactly two childrenhave exactly two children

Balanced binary tree: for each node, theBalanced binary tree: for each node, the

difference in depth of the right and left subtrees isdifference in depth of the right and left subtrees isno more than oneno more than one

Completely balanced tree: left and right subtreesCompletely balanced tree: left and right subtrees

of every node have the same heightof every node have the same height

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Binary Trees StorageBinary Trees Storage

Linked List based implementationLinked List based implementation

Array based implementationArray based implementation

The tree is a conceptual structureThe tree is a conceptual structure The data can be stored either in aThe data can be stored either in a dynamic linked treedynamic linked tree

structure, or in contiguous memory cells (array) according to astructure, or in contiguous memory cells (array) according to a

set patternset pattern; in other words, implementation can be pointer; in other words, implementation can be pointer--

based or arraybased or array--basedbased

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Binary Tree: as a linkedBinary Tree: as a linked

structurestructure Each node in the tree consists of:Each node in the tree consists of:

The data, or value contained in the elementThe data, or value contained in the element

A left child pointer (pointer to first child)A left child pointer (pointer to first child)

A right child pointer (pointer to second child)A right child pointer (pointer to second child)

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Binary Tree StructureBinary Tree Structure

The representation of a binary tree structure is relativelyThe representation of a binary tree structure is relativelystraight forward.straight forward.

We need a variable to store the data at the node and 2We need a variable to store the data at the node and 2

poin

ters to the left an

d right subtrees.poin

ters to the left an

d right subtrees.

struct Node {struct Node {int dataint dataNode *leftNode *leftN

ode *rightN

ode *right}}

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Binary Tree StructureBinary Tree Structure

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Binary Tree: contiguous storageBinary Tree: contiguous storage

Value in root node stored first, followed by left child, thenValue in root node stored first, followed by left child, thenright childright child

Each successive level in the tree stored left to right; unusedEach successive level in the tree stored left to right; unusednodes in tree represented by a bit pattern to indicate nothingnodes in tree represented by a bit pattern to indicate nothing

stored therestored there Children of any givennodeChildren of any givennode nn is stored in cellsis stored in cells 2n2n andand 2n + 12n + 1

(If array index starts at 1)(If array index starts at 1)

Can calculate position for any givennodeCan calculate position for any givennode

Storage allocated as for full tree, even if many nodes emptyStorage allocated as for full tree, even if many nodes empty

For a tree ofFor a tree ofdepthdepth hh we need array of 2we need array of 2hh+1+1

--1 cells1 cells

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Common OperationsCommon Operations

Tree traversalTree traversal

Node additionNode addition

Node deletionNode deletion

DestroyDestroy

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Binary Tree TraversalsBinary Tree Traversals AAbinary tree traversalbinary tree traversalrequires that each node of the tree berequires that each node of the tree be

processed once and only once in a predetermined sequence.processed once and only once in a predetermined sequence.

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Traversal of Binary TreesTraversal of Binary Trees

Pass through all nodes of treePass through all nodes of tree

Inorder (symmetric traversal)Inorder (symmetric traversal)

Preorder (depth first traversal)Preorder (depth first traversal)

PostorderPostorder

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Tree Traversal TypesTree Traversal Types

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Preorder TraversalPreorder Traversal

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Preorder TraversalPreorder Traversal

Root Left Right mannerRoot Left Right manner

++ LeftLeft RightRight

++ [*Left Right][*Left Right] [+Left Right][+Left Right]

++ (*AB)(*AB) [+ *Left Right E][+ *Left Right E] +*AB+*AB+ *C D E+ *C D E

+

* +

A B * E

C D

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Inorder TraversalInorder Traversal

In the inorder traversal, the left subtree is processed first,In the inorder traversal, the left subtree is processed first,

followed by the root node, and then the right subtree.followed by the root node, and then the right subtree.

Inorder = root node in betweenthe left and right subtrees.

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In order TraversalIn order Traversal

In an inorder traversal a node is visited after its left subtree and

before its right Subtree

Application: draw a binary tree orArithmetic expression

prin

tin

g

((2 (a 1)) + (3 b))

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Inorder TraversalInorder Traversal

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Example of Binary TreeExample of Binary Tree

(inorder)(inorder)

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Post Order TraversalPost Order Traversal

In a postorder traversal, a node is visited after its descendants

Application: compute space used by files in a directory and its

subdirectories

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Postorder TraversalPostorder Traversal

In the postorder traversal, the left subtree is processedIn the postorder traversal, the left subtree is processed

first, followed by the right subtree, and then the rootfirst, followed by the right subtree, and then the root

node.node.

Postorder = root node afterthe left and right subtrees.

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Left Right Root mannerLeft Right Root manner

LeftLeft RightRight ++

[Left Right *][Left Right *] [Left Right+] +[Left Right+] +

(AB*)(AB*) [Left Right * E + ]+[Left Right * E + ]+

(AB*)(AB*) [C D * E + ]+[C D * E + ]+

AB*AB* C D * E + +C D * E + +

+

* +

A B * E

C D

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Finite set ofnodes that is either empty, or consists of a root

and two disjoint binary trees called the left subtree and right

subtree.

Node contains information and two pointers to othernodes

Each node has at most two children

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Visiting and Traversing a NodeVisiting and Traversing a Node

Many applications require that all of the nodes of a

tree be visited.

Visiting a node may mean printing contents,

retrieving information, making a calculation, etc.

Traverse: To visit all the nodes in a tree in a

systematic fashion.

A traversal can pass through a node without visiting it at

that moment.

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