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Data Structures: Binary Trees

Data Structures: Binary Trees. CMSC 11500 Introduction to Computer Programming October 28, 2002. Roadmap. Recap: Family trees Generalizing: Binary trees Data definition Template Functions on Binary Trees Contains? Map-tree Fringe Summary. Family Trees. Data definition:

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Data Structures: Binary Trees

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  1. Data Structures:Binary Trees CMSC 11500 Introduction to Computer Programming October 28, 2002

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

  3. 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

  4. Binary Trees • More general form • Nodes with two self-references • Family trees -> self-refs= mother, father • Data definition: • (define-struct bt (val left right)) • Where val:number, left, right are binary-tree • Binary-tree is: • #f, or • (make-bt val left right)

  5. 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))

  6. Binary Tree Graphic 123 12 3 1 2

  7. 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))….))))))

  8. 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

  9. Contains? • Contract: binary-tree -> boolean • Purpose: To determine if a given value is in the tree

  10. 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))))))))

  11. Map-tree • Analogous to map for flat lists • Applies some function to every element • Contract:map-tree • (number->number) binary-tree -> binary-tree • Purpose: • To apply function to every element of tree

  12. 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)))))

  13. Using Map-tree • Square-tree: • Contract: square-tree: binary-tree -> binary-tree • Purpose: Square all numbers in tree • Double-tree: • Contract: double-tree: binary-tree -> binary-tree • Purpose: Double all numbers in tree

  14. Using Map-tree (define (square-tree abt) (map-tree square abt)) (define (double-tree abt) (map-tree (lambda (x) (+ x x)) abt))

  15. Next Time • Binary Search Trees • Invariants • Efficient set implementation

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