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:– (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

123

12 3

1 2

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