Data Structures: Binary Trees CMSC 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: –(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

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)

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))

Binary Tree Graphic

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))….))))))

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

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

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))))))))

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

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)))))

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

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

Next Time Binary Search Trees –Invariants –Efficient set implementation