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Discrete StructuresTrees (Ch. 11)

Dr. Muhammad HumayounAssistant Professor

COMSATS Institute of Computer Science, Lahore.mhumayoun@ciitlahore.edu.pk

https://sites.google.com/a/ciitlahore.edu.pk/dstruct/Modified slides of Dr. M. Atif

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First we need

some definitions from Graph theory

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Path

• A path is a sequence of edges that begins at a vertex of a graph and travels from vertex to vertex along edges of the graph.

• Formal Definition: Let n Z+ ∈ and G an undirected graph. A path of length n from vertices u to v in G is a sequence of n edges e1 , ..., en of G for which there exists a sequence x0=u, x1, . . . , xn−1, xn=v of vertices.

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Example

• Find a path between a and e.• a, d, c, f , e is a simple path of length 4, because {a, d}, {d, c},

{c, f }, and {f, e} are all edges.• a, e is a simple path of length 1, because {a, e} is the edge

connecting both vertices.• a, b, f , e is a simple path of length 3, because {a, b}, {b, f},

and {f, e} are all edges.• …

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Circuit

• The path is a circuit if it begins and ends at the same vertex, that is, if u = v, and has length greater than zero.

Path a, b, c, d, a is a circuit.

A path or circuit is simple if it does not contain the same edge more than once.

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Connected Graph

• An undirected graph is called connected if there is a path between every pair of distinct vertices of the graph.

• The graph G1 is connected.• The graph G2 is not

connected. (For instance, there is no path in G2 between vertices a and d.)

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Trees• A (free) tree is an undirected graph T such that– T has no simple circuits– T is connected

• A tree cannot contain multiple edges or loops. Therefore any tree must be a simple graph.

• Theorem: An undirected graph is a tree iff there is a unique simple path between any two of its vertices.

Tree

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• Which of the graphs are trees?• G1 and G2 are trees, because both are connected

graphs with no simple circuits.• G3 is not a tree because e, b, a, d, e is a simple circuit

in this graph. • G4 is not a tree because it is not connected.

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Forest

• A forest is an undirected graph such that– It has no simple circuits– It is not necessarily connected

• The connected components of a forest are trees

ForestOne graph with 3 trees

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Rooted Tree• A rooted tree is a tree in which one vertex has

been designated as the root and every edge is directed away from the root.

• Root, parent, child, siblings relations between vertices

Tree and Rooted Tree

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SubtreeIf a is a vertex in a tree, the subtree with a as its root is the subgraph of the tree consisting of a and its descendants and all edges connected to these descendants.

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Spanning Tree

• A spanning tree of a connected graph is a spanning subgraph that is a tree

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Spanning tree of the simple graph

• it is not a tree because it contains three simple circuits.

• Which ones?– {a, e}, {e,f} and {c,g}

• Spanning tree:• Can we produce more

spanning trees?

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Some facts

• A simple graph is connected if and only if it has a spanning tree.

• A spanning tree is not unique unless the graph is a tree

• Spanning trees have applications to the design of communication networks

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Tree Traversal

• Ordered rooted trees are often used to store information.

• We need procedures for visiting each vertex of an ordered rooted tree to access data.

• We will study some important algorithms for visiting all the vertices of an ordered rooted tree.

• Ordered rooted trees can also be used to represent various types of expressions.

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Ordered Rooted Tree• An ordered rooted tree is a rooted tree where

the children of each internal vertex are ordered.

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Traversal Algorithms

• Procedures for systematically visiting every vertex of an ordered rooted tree are called traversal algorithms.

• Most commonly used algorithms:–Preorder traversal – Inorder traversal–Postorder traversal

• All recursively defined.

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Preorder Traversal

• Let T be an ordered rooted tree with root r. If T consists only of r, then r is the preorder traversal of T .

• Otherwise, suppose that T1, T2, . . . , Tn are the subtrees at r from left to right in T. The preorder traversal begins by visiting r.

• It continues by traversing T1 in preorder, then T2 in preorder, and so on, until Tn is traversed in preorder

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Procedureprocedure preorder(T:ordered rooted tree)r:= root of TList/print rfor each child e of r from left to right T(e) := subtree with e as its root preorder(T(e))

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Inorder Traversal

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Inorder traversal

• Let T be an ordered rooted tree with root r. If T consists only of r, then r is the inorder traversal of T .

• Otherwise, suppose that T1, T2, . . . , Tn are the subtrees at r from left to right.

• The inorder traversal begins by traversing T1 in inorder, then visiting r. It continues by traversing T2 in inorder, then T3 in inorder, . . . , and finally Tn in inorder.

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Procedureprocedure inorder (T : ordered rooted tree)r := root of Tif r is a leaf then list/print relse l := first child of r from left to right T (l) := subtree with l as its root inorder(T (l)) list/print r for each child c of r except for l from left to right T (c) := subtree with c as its root inorder(T (c))

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Postorder Traversal

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Postorder Traversal

• Let T be an ordered rooted tree with root r. If T consists only of r, then r is the postorder traversal of T .

• Otherwise, suppose that T1, T2, . . . , Tn are the subtrees at r from left to right. The postorder traversal begins by traversing T1 in postorder, then T2 in postorder, . . . , then Tn in postorder, and ends by visiting r.

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Proceedure

procedure postorder(T : ordered rooted tree)r := root of Tfor each child c of r from left to right T (c) := subtree with c as its root postorder(T (c))list/print r

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Infix, Prefix, and Postfix Notation

• We can represent complicated expressions, such as compound propositions, combinations of sets, and arithmetic expressions using ordered rooted trees.

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Ordered rooted tree of an Expression

• ((x + y) ↑ 2) + ((x − 4)/3)

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Infix notation

• It is the common arithmetic and logical formula notation, in which operators are written between the operands they act on (e.g. 2 + 2).

• Inorder traversal.

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Prefix form

• Also known as Polish notation.• A form of notation for logic, arithmetic, and

algebra.• It places operators to the left of their operands.• (5 − 6) * 7 Infix notation• * - 5 6 7 Prefix notation• 5 − (6 * 7) Infix notation• − 5 * 6 7 Prefix notation

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Prefix form

• What is prefix form of ((x + y) ↑ 2) + ((x − 4)/3)?• + ↑ + x y 2 / − x 4 3.

• Preorder traversal

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Evaluating a prefix form

What is the value of the prefix expression + − 2 3 5∗ /↑ 2 3 4?

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Postfix form

• Also known as Reverse Polish Notation.• We obtain the postfix form of an expression by

traversing its tree in postorder.

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Example

What is the postfix form of the expression: ((x + y) ↑ 2) + ((x - 4)/3) ?

Postorder Traversal

x y + 2 ↑ x 4 - 3 / +

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Evaluating a Postfix Expression

Postfix expression: 7 2 3 * - 4 t 9 3/+

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Representing other types of expressions with Rooted trees

Compound proposition: (￢ (p q)) ↔ (∧ ￢ p ∨￢ q).

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Binary Trees• Binary rooted Trees• A binary tree is a tree data structure in which

each node has at most two child nodes, usually distinguished as "left" and "right".

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END