Trees I think that I shall never see Trees I A poem lovely as a - - PDF document

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Trees I Trees

I think that I shall never see A poem lovely as a tree

• - J. Kilmer

Trees

• In CS, we look at trees

from the bottom up

Anatomy of a Tree

• Node

– Basic element of the tree – Contains a piece of information – Represented by the circles a b c d e f g h i j

Anatomy of a Tree

• Child

– Direct descendant of a node – Node a has 2 children: b and c – A node may have multiple children

• Parent

– Direct ancestor of a node – The parent of node b and c are Node a – A node can have only 1 parent

a b c d e f g h i j

Anatomy of a Tree

• Sibling

– Nodes that share the same parent – Nodes b and c are siblings

• Leaf

– Node that has no children – Nodes d,e,h,i, and j are leaves a b c d e f g h i j

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Anatomy of a Tree

• Root

– Topmost node of the tree – The root is an orphan: It has no parent. – Node a is the root of this tree. a b c d e f g h i j

Anatomy of a Tree

• Root

– Topmost node of the tree – The root is an orphan: It has no parent. – Node a is the root of this tree. a b c d e f g h i j

Anatomy of a Tree

• Trees are strictly

acyclic

– A child node cannot be a parent to one of its ancestors – This figure is not a valid tree – Node c has 2 parents! a b c d e f g h i j

Anatomy of a Tree

• Subtree

– All children of a node, can be considered the root of it’s own tree – These are called subtrees – I smell recursion! a b c d e f g h i j

Anatomy of a Tree

• Balanced Tree

– A tree is balanced if all child subtrees have similar depths – This tree is not balanced since Node c’s subtree has a depth

• f 2 but Node b’s

subtree has a depth of 1. a b c d e f g h i j

Anatomy of a Tree

• Binary Tree

– A binary tree is a tree where each node has at most 2 children. – This tree is not a binary tree a b c d e f g h i j

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Anatomy of a Tree

• Binary Tree

– But this tree is – Children of nodes in a binary tree are referred to as left child or right child

• Node h is the left child
• f Node f
• Node i is the right child
• f Node f

a b c d e f g h i

Anatomy of a Tree

• Full Binary Tree

– A binary tree is full if

• All of it’s leaf nodes are
• f the same depth
• Each non-leaf node has

2 children

– This binary tree is not full. a b c d e f g h i

Anatomy of a Tree

• Full Binary Tree

– But this one is. a b c d e f g

Anatomy of a Tree

• Complete Binary Tree

– A binary tree is complete if

• Each level (except the

deepest) must contain as many nodes as possible

• At the deepest level, all

nodes as as far left as possible

– This binary tree is not complete. a b c d e f g h i

Anatomy of a Tree

• Complete Binary Tree

– But this one is!

• Questions?

a b c d e f g h i

Traversing a Tree

• A means to process all the nodes in a tree

– A traversal starts at the root – Visits each node exactly once

• “Processes” the data in a node when visited

– Nodes can be visited in different orders

• Breadth-first traversal
• Depth-first traversal

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Anatomy of a Tree

• Breadth-first

– All the nodes at a given level are visited before the nodes at the next level – Example:

• a,b,c,d,e,f,g,h,i

a b c d e f g h i

Anatomy of a Tree

• Depth-first

– At each node, the node is visited as well as it’s children’s subtrees. – Different types based

• n order
• Pre-order
• In-order
• Post-order

a b c d e f g h i

Anatomy of a Tree

• Pre-order

– At each node

• The node is visited first
• Pre-order traversal of

the left subtree

• Pre-order traversal of

the right subtree

– Example:

• a,b,d,h,i,e,c,f,g

a b c d e f g h i

Anatomy of a Tree

• in-order

– At each node

• In-order traversal of the

left subtree

• The node is visited next
• The In-order traversal
• f the right subtree

– Example:

• h,d,i,b,e,a,f,c,g

a b c d e f g h i

Anatomy of a Tree

• post-order

– At each node

• Post-order traversal of

the left subtree

• Post-order traversal of

the right subtree

• The node is visited next

– Example:

• h,i,d,e,b,f,g,c,a

a b c d e f g h i

What are trees good for?

• Hierarchical relationships

Performer Actor Musician isA isA Guitarist Pianist Drummer

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What are trees good for?

• Binary trees can be used to represent

decision taxonomies

Mammal? Bigger than a cat? Underwater? Trout Bird Elephant Mouse

yes yes yes no no no

What are trees good for?

• Branching can also imply ordering of node

data

45 30 55 50 60 22 35

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Summary

• Trees

– Terminology

• Root
• Children
• Leaf

– Trees are acyclic – Traversal

• Breadth-first
• Depth-first

– Pre-order – In-order – Post-Order

Next Time

• Coding a binary tree in Java