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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
7 8
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
7 8
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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2 3 4
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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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2 3 4
5 6 87
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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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2 3 4
5 6 87
9 10 1211
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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2 3 4
5 6 87
9 10 1211
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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3
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4
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
1root
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2 3
77 88
3 44
1
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
1root
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2
55 66
2 3
77 88
3 44
1
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
