trees : part 1 section 4.1 (1) theory and terminology (2) preorder, postorder and levelorder...

Trees : Part 1

Section 4.1

(1) Theory and Terminology(2) Preorder, Postorder and Levelorder Traversals

Theory and Terminology

Definition: A tree is a connected graph with no cycles

Consequences: Between any two vertices, there is exactly one

unique path

A Tree?

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A Tree?

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Theory and Terminology

Definition: A rooted tree is a graph G such that: G is connected G has no cycles G has exactly one vertex called the root of the

tree

Theory and Terminology

Consequences The depth of a vertex v is the length of the

unique path from root to v G can be arranged so that the root is at the top,

its neighboring vertices are vertices of depth 1, and so on…

The set of all vertices of depth k is called level k of the tree

A Rooted Tree

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root

depth = 1

depth = 0

depth = 3

depth = 2

height = 2

height = 3

height = 0

height = 1

Rooted Tree: Recursive definition

A graph with N nodes and N - 1 edges Graph has

one root r Zero or more non-empty sub-trees, each of whose root is

connected to r by an edge.

Every node except the root has one parent

Theory and Terminology

Definition: A descending path in a rooted tree is a path, whose edges go from a vertex to a deeper vertex

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root

depth = 1

depth = 0

depth = 3

depth = 2

3

8

12

Theory and Terminology Consequences:

A unique path from the root to any vertex is a descending path

The length of this path is the depth of the vertex

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root

depth = 1

depth = 0

depth = 3

depth = 2

1

3

8

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Theory and Terminology

Definition: If there is a descending path from v1 to v2, v1 is an ancestor of v2, and v2 is a descendant of v1.

Theory and Terminology

Suppose v is a vertex of depth k: Any vertex that is adjacent to v must have

depth k - 1 or k + 1.

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root

depth = 1

depth = 0

depth = 3

depth = 2

3

1

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Theory and Terminology

Suppose v is a vertex of depth k: Vertices adjacent to v of depth k + 1 are

called children of v.

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root

depth = 1

depth = 0

depth = 3

depth = 2

3

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Theory and Terminology

Suppose v is a vertex of depth k: If k > 0, there is exactly one vertex of depth k – 1

that is adjacent to v in the graph. This vertex is called the parent of v.

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root

depth = 1

depth = 0

depth = 3

depth = 2

3

1

Theory and Terminology

Definitions A vertex with no children is called a leaf

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root

depth = 1

depth = 0

depth = 3

depth = 2

Theory and Terminology Definitions

Depth of a vertex v is its distance from the root. Height of a vertex v is the distance of the longest path from

v to one of its descendant leaves. The height of a tree is the maximum depth of its vertices

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root

depth = 1

depth = 0

depth = 3

depth = 2

height

Theory and Terminology

Definitions The root is the only vertex of depth 0. The root has no

parent.

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root

depth = 1

depth = 0

depth = 3

depth = 2

1

Example of rooted tree

Which are the parent nodes? Which are the child nodes? Which are the leaves? What is the height and depth of the tree? What is the height and depth of node E? Node F?

Overview of Tree Implementation

Each node points to Its first child Its next sibling Back to its parent (optional)

What could be an alternate representation?

Tree Traversals

Definition: A traversal is the process for “visiting” all of the vertices in a tree Often defined recursively Each kind corresponds to an iterator type Iterators are implemented non-recursively

Preorder Traversal

Visit vertex, then visit child vertices (recursive definition)

Depth-first search Begin at root Visit vertex on arrivalarrival

Implementation may be recursive, stack-based, or nested loop

Preorder Traversal

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root

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Preorder Traversal of UNIX Directory Tree

Postorder Traversal

Visit child vertices, then visit vertex (recursive definition)

Depth-first search Begin at root Visit vertex on departuredeparture

Implementation may be recursive, stack-based, or nested loop

Postorder Traversal

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Postorder TraversalCalculating Size of Directory

Levelorder Traversal

Visit all vertices in level, starting with level 0 and increasing

Breadth-first search Begin at root Visit vertex on departure

Only practical implementation is queue-based

Levelorder Traversal

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3 44

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Tree Traversals

Preorder: depth-first search (possibly stack-based), visit on arrival

Postorder: depth-first search (possibly stack-based), visit on departure

Levelorder: breadth-first search (queue-based), visit on departure