Discrete Structures Trees (Ch. 11) Dr. Muhammad Humayoun Assistant 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

First we need some definitions from Graph theory

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.

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. …

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.

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.)

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

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.

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 Forest One graph with 3 trees

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

Subtree If 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.

Spanning Tree A spanning tree of a connected graph is a spanning subgraph that is a tree

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?

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

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.

Ordered Rooted Tree An ordered rooted tree is a rooted tree where the children of each internal vertex are ordered.

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.

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

Procedure procedure preorder(T:ordered rooted tree) r:= root of T List/print r for each child e of r from left to right T(e) := subtree with e as its root preorder(T(e))

Inorder Traversal

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.

Procedure procedure inorder (T : ordered rooted tree) r := root of T if r is a leaf then list/print r else 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))

Postorder Traversal

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.

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

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

Ordered rooted tree of an Expression ((x + y) ↑ 2) + ((x − 4)/3)

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.

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

Prefix form What is prefix form of ((x + y) ↑ 2) + ((x − 4)/3)? Preorder traversal

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

Postfix form Also known as Reverse Polish Notation. We obtain the postfix form of an expression by traversing its tree in postorder.

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

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

Representing other types of expressions with Rooted trees Compound proposition: (￢(p ∧ q)) ↔ (￢p ∨￢q).

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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