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Foundations of Software DesignFall 2002Marti Hearst

Lecture 15: Trees 

 

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Trees

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Trees• Trees are very important and useful• They are usually drawn upside-down• The allow us to represent hierarchy

– File system– Book structure– Employees in a bureacracy

• Every node has exactly one parent – Except the root

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Anatomy of a Tree

root

siblings

parent &child

Ancestor & descendent

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Tree Terminology• Parent / Child

– If node u is the parent of node v, then v is the child of u• Siblings

– Two nodes that are children of the same parent.• Leaf (External node)

– A node is a leaf if it has no children• Internal node

– A node is internal if it has one or more children• Ancestor / Descendent

– An ancestor of a node is either the node itself or an ancestor of the parent of the node

– (this is recursive)• Subtree

– The subtree of T rooted at node v is the tree consisting of all the descendents of v in T (including v)

Adapted from http://www.oopweb.com/Algorithms/Documents/PLDS210/VolumeFrames.html

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Binary Tree• The simplest form of tree is a Binary Tree• A Binary Tree consists of

– (a) A node (called the root node) and– (b) Left and right subtrees– Both the subtrees are themselves binary trees

• Note: this is a recursive definition

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Binary Tree

Adapted from http://www.oopweb.com/Algorithms/Documents/PLDS210/VolumeFrames.html

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Tree Terminology• Proper tree

– A binary tree is proper if every node has either 2 or 0 children

– Another way to say this: • all internal nodes have exactly 2 children

– Even an improper binary tree is still a tree• Complete tree

– A tree (binary or otherwise) is complete if every leaf is the same distance from the root

Proper treeComplete tree

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Trees vs. Graphs / Networks• The Web is not a tree. Why not?

– Nodes in a tree can have only one parent• Graphs / Networks are the more general case

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Tree TerminologyHeight

– The height of a tree is the height of the root• Intuitively – the length of the longest path from the root

to a leaf, across all the leaves– More formally, the height of a node v is

• 0 if v is a leaf• 1 + maximum height of v’s children

– (this is a recursive definition)

Adapted from http://www.oopweb.com/Algorithms/Documents/PLDS210/VolumeFrames.html

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Tree TerminologyDepth

– The depth of a node v in T is the number of ancestors of v, excluding v itself.

– (The inverse of height, from the node’s viewpoint)– More formally:

• If v is the root, the depth of v is 0• Otherwise, the depth of v is one plus the depth of the

parent of v

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Our Binary Tree• Defined recursively as consisting of

– Data– A lefthand Binary Tree– A righthand Binary Tree

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Simple Binary Tree Code

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Recursion vs. Iterationprinting the left side of the tree

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Tree Traversal• Preorder• Inorder• Postorder

• All defined in terms of – When do you visit the node itself– When do you visit the lefthand side– When do you visit the righthand side

root

aaa

abaa

a

aab

b

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PreOrder root

aaa

abaa

a

aab

b

Why? Textual representation of the tree (parents before children)

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InOrderroot

aaa

abaa

a

aab

b

Why? Useful for searching in ordered trees

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PostOrderroot

aaa

abaa

a

aab

b

Why? Computing a property of a node depends on the nodes below it.

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Depth-first Search

From http://www.rci.rutgers.edu/~cfs/472_html/AI_SEARCH/ExhaustiveSearch.html

Full tree

DFS (to see it, run inPresentationMode)

Note: DFS traversal is equivalent to PreOrder traversal

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Breadth-first Search

From http://www.rci.rutgers.edu/~cfs/472_html/AI_SEARCH/ExhaustiveSearch.html

Full tree

BFS (to see it, run inPresentationMode)

What do we need besides the runtime stack to make BFS work?

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Breadth First Traversal

We can reuse our Queue!!

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Depth First Traversal

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Depth First Search

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Algorithm Analysis• What is the running time for

– Depth First Traversal?• O(n)

– Depth First Search?• Worst case: O(n)

– Breadth First Traversal?• O(n)

– Breadth First Search• Worst case: O(n)

– How could we improve these search times?