Data Structures: Binary Trees

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Data StructuresBinary Trees

• CMSC 11500
• Introduction to Computer Programming
• October 28, 2002

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Roadmap

• Recap Family trees
• Generalizing Binary trees
• Data definition
• Template
• Functions on Binary Trees
• Contains?
• Map-tree
• Fringe
• Summary

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Family Trees

• Data definition
• (define-stuct ft (name eye-color mother father))
• Where name, eye-color symbols, mother, father
  are
• Family-tree is
• 1) unknown, or
• 2) (make-ft name eye-color mother father)
• (define nana (make-ft alice blue unknown
  unknown))
• (define ma (make-ft anna brown nana pap))
• Functions
• Blue-eyed ancestor, all-blue-eyed-ancestors,
  how-deep

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Binary Trees

• More general form
• Nodes with two self-references
• Family trees -gt self-refs mother, father
• Data definition
• (define-struct bt (val left right))
• Where valnumber, left, right are binary-tree
• Binary-tree is
• f, or
• (make-bt val left right)

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Binary Tree Examples

• Vocabulary
• If bt-left bt-right both f, leaf node
• If not bt-left or bt-right of other node, root
• (define leaf1 (make-bt 1 f f))
• (define leaf2 (make-bt 2 f f))
• (define leaf3 (make-bt 3 f f))
• (define mid12 (make-bt 12 leaf1 leaf2))
• (define root (make-bt 123 mid12 leaf3))

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Binary Tree Graphic
123
12
3
1
2
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Binary Tree Template

• Questions
• How many conditions? , How many parts?, How many
  what self-references?
• (define (fn-for-btree abt)
• (cond ((eq? abt f) .)
• ((bt? abt)
• (cond ((eq? (bt-val abt) ) .)
• . (fn-for-btree (bt-left
  abt))
• . (fn-for-btree (bt-right
  abt)).))))))

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Functions on Binary Trees

• Is x in the tree?
• Contains?
• Do something to every element in the tree
• map-tree
• Find the nodes with no successors
• fringe

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Contains?

• Contract binary-tree -gt boolean
• Purpose To determine if a given value is in the
  tree

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Contains?
(define (contains? testnum abt)
(cond ((eq? abt f) f)
((bt? abt)
(cond ((eq? (bt-val abt) testnum) t)
(else (or (contains? testnum (bt-left abt))
(contains? testnum (bt-right abt))))))))
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Map-tree

• Analogous to map for flat lists
• Applies some function to every element
• Contractmap-tree
• (number-gtnumber) binary-tree -gt binary-tree
• Purpose
• To apply function to every element of tree

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Map-tree
(define (map-tree func abt) (cond ((eq? abt f)
f) ((bt? abt) (make-bt (func (bt-val
abt)) (map-tree func (bt-left
abt)) (map-tree func (bt-right abt)))))
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Using Map-tree

• Square-tree
• Contract square-tree binary-tree -gt binary-tree
• Purpose Square all numbers in tree
• Double-tree
• Contract double-tree binary-tree -gt binary-tree
• Purpose Double all numbers in tree

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Using Map-tree
(define (square-tree abt) (map-tree square
abt)) (define (double-tree abt) (map-tree
(lambda (x) ( x x)) abt))
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Next Time

• Binary Search Trees
• Invariants
• Efficient set implementation