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FOA

Lecture 10Trees

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TreesImposes a Hierarchical structure, on a collection of items.

Applications can be: Organization charts. Organization of information in database systems. Representation of syntactic structure of source

programs in compilers.

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TreesPakistan BOI Organizational chart

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TreesRoyal Genealogies

Roman Emperors: Julian-Claudian House (27BC-68AD)Roman Emperors: Julian-Claudian House (27BC-68AD) France

CHRONOLOGY The Merovingian Kings (481-752) Pepin and Charlemagne (640-987)

Monarchs of France (987-1883) Capet to Valois (987-1328)

House of Valois (1328-1589) the French Bourbons 1589-1883)

                                 

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Trees

Organization of information in database systems.

CUSTOMERCUSTOMER NUMBERCUSTOMER NAMECUSTOMER CITYCUSTOMER POSTCUSTOMER STCUSTOMER ADDRCUSTOMER PHONECUSTOMER FAX

ORDERORDER NUMBERORDER DATESTATUS

ORDER ITEM BACKORDEREDQUANTITY

ITEMITEM NUMBERQUANTITYDESCRIPTION

ORDER ITEM SHIPPEDQUANTITYSHIP DATE

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Trees: Basic TerminologyA Tree is a collection of elements called nodes.

One of the node is distinguished as a root, along with a relation (“parenthood”) that places a hierarchical structure on the nodes.

A node can be of any type. e.g.; A Letter, a String, or a Number.

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TreesA Tree can be defined recursively.

A single node by itself is a tree. This node is also the root of the tree.

Suppose n is a node and T1,T2,…Tk are trees with roots n1,n2,…nk, respectively.

Construct a new tree by making n the parent of nodes n1,n2,…nk.

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Trees

T1,T2,…Tk the subtrees.

n1,n2,…nk the children of node n.

Null Tree: A Tree with no nodes is called a “Null Tree”.

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Table of Contents: An Example of Trees

BookC1

s1.1s1.2

C2s2.1

s2.1.1s2.1.2

s2.2s2.3

C3

Book

C1 C2 C3

s1.1 s1.2 s2.1 s2.2 s2.3

s2.1.1 s2.1.2

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The root node, called “Book”, has three subtrees.

The root of each subtree corresponds to a chapter, C1, C2, C3

The downward line represents: “Book” as the parent of C1, C2, C3 and these are the children of “Book”

Table of Contents: An Example of Trees

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The subtree, with root C3 is a tree of single node.

The subtree with root C2 has three subtrees, corresponding to the subsections s2.1, s2.2 and s2.3.

Table of Contents: An Example of Trees

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Trees: Some Definitions Path: If n1,n2,…nk is a sequence of nodes in a tree

such that ni is a parent of ni+1 for i <= I < k, then this sequence is called a Path from n1 node to nk.

Length of Path: One less than the number of nodes in the path.

There is a path of length zero from every node to itself.

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Zero length pathBook

C1 C2 C3

s1.1 s1.2 s2.1 s2.2 s2.3

s2.1.1 s2.1.2

Trees: Some Definitions

Path from C2 to s2.1.1Path length is 2

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Trees: Some Definitions If there is a path from node a to node b, then a is an

ancestor of b and b is a descendant of a.

Any node is both an ancestor and descendant of itself.

An ancestor or descendant of a node, other than the node itself, is called a proper ancestor or proper descendant of the node.

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Book

C1 C2 C3

s1.1 s1.2 s2.1 s2.2 s2.3

s2.1.1 s2.1.2

Trees: Some Definitions

C2 ancestor of s2.1s2.1 ancestor of s2.1.1

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Trees: Some Definitions The root is the only node with no proper ancestor.

A node with no proper descendant is called a leaf.

A subtree of a node is a node, together with all its descendants.

The height of a node in a tree is the length of a longest path from that node to the leaf.

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Book

C1 C2 C3

s1.1 s1.2 s2.1 s2.2 s2.3

s2.1.1 s2.1.2

Trees: Some DefinitionsNo proper ancestor

No proper descendent

Sub tree

Height

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Trees: Some Definitions The height of a tree is the height of the

root.

The depth of a node is the length of the path from the root to that node.

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Book

C1 C2 C3

s1.1 s1.2 s2.1 s2.2 s2.3

s2.1.1 s2.1.2

Trees: Height of a Tree

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The Order of NodesThe children of a node are usually ordered from left-to-right.

b c

a

c b

a

Two distinct (ordered) trees

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The Order of NodesComparison on bases of ordering:

“If a and b are siblings, and a is to the left of b, then all the descendants of a are to the left of all the descendent of b.”

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The Order of Nodes

1

2 3 4

5 6 7

8 9 10

Node 8 is to the right of node 2.

Node 8 is the left of nodes 9,6,10,4 and 7.

Neither left nor right of its ancestor 1,3 and 5.

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The Order of NodesThree most important ordering are: Preorder

Postorder

InorderThese ordering are defined recursively.

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The Order of Nodes If a tree T is null, then the empty list is

the preorder, inorder and postorder listing of T.

If T consists a single node, then that node by itself is the preorder, inorder and postorder listing of T.

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The Order of NodesLet T be a tree with root n and subtrees

T1,T2,…Tk

n

T1 T2 Tk…

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The Order of NodesThe preorder listing of the nodes of T is the root n of T followed by the nodes of T1 in preorder, then the nodes of T2 in preorder, and so on, up to the nodes of Tk in preorder.

n

T1 T2 Tk…

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The Order of NodesThe inorder listing of the nodes of T is the nodes of T1 in inorder, followed by the node n, followed by the nodes of T2,…, Tk, each group of nodes in inorder.

n

T1 T2 Tk…

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The Order of NodesThe postorder listing of the nodes of T is the nodes of T1 in postorder, then the nodes of T2 in postorder, and so on, up to Tk, all followed by node n.

n

T1 T2 Tk…

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1

2 3 4

5 6 7

8 9 10

The Order of NodesWalk around the outside of the tree, start at the root, move counterclockwise and stay as close to the tree as is possible.

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The Order of Nodes For preorder, list the nodes the first time

you pass it. For postorder, list a nodes the last time

you pass it, as you move up to the parent.

For inorder, list a leaf the first time you pass it, but list an interior node the second time you pass it.

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The Order of Nodes

The ordering of the interior nodes and their relationship to the leaves vary among the three orderings.

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Labeled and Expressions TreesA label is a value stored at the node.

n4 n5

n2

n6 n7

n3

n1*

+ +

a ab c

A labeled tree representing expression (a + b) * ( a + c)

LabelName ofThe node

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Labeled and Expressions Trees1. Every leaf is labeled by an operand and

consists of that operand alone.2. Every interior node n is labeled by an

operator. If n represents the binary operator Θ, the left child represent the expression E1 and the right child represent the Expression E2, then n represents (E1) Θ (E2).

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Labeled and Expressions Trees The Preorder listing of the Expression tree produces

a prefix expression. In a prefix expression the operator precedes its left

and right operand. The prefix operation for a single operand a is a itself. The prefix expression for (E1) Θ (E2), is Θ P1 P2,

where P1 and P2 are the prefix expressions for E1 and E2.

No parentheses are necessary.

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Labeled and Expressions Trees

n4 n5

n2

n6 n7

n3

n1

+ +

a ab c

The preorder listing of the labels is *+ab+ac *

The prefix expression for n2 is +ab, is the shortest legal prefix of *+ab+ac.

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Labeled and Expressions Trees The Postorder listing of the Expression tree

produces a postfix (or Polish)expression. In a postfix expression the operator follows

its left and right operand. The postfix expression for (E1) Θ (E2), is P1

P2 Θ, where P1 and P2 are the postfix expressions for E1 and E2.

No parentheses are necessary.

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Labeled and Expressions Trees

n4 n5

n2

n6 n7

n3

n1

+ +

a ab c

The postorder listing of the labels is ab+ac+* *

The postfix expression for n2 is ab+, is the shortest legal postfix of ab+ac+*.

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Labeled and Expressions Trees

n4 n5

n2

n6 n7

n3

n1

+ +

a ab c

The inorder traversal gives the infix expression, but without any parentheses.

a+b*a+c