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前缀、中缀、后缀表达式(转载)
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關鍵字： 概念, 前綴表達式, 前綴記法, 中綴表達式, 中綴記法, 波蘭式, 后綴表達式, 后綴記法, 逆波蘭式
中綴表達式（中綴記法） 前綴表達式（前綴記法、波蘭式） 前綴表達式的計算機求值： 將中綴表達式轉換為前綴表達式：
掃描到的元素 S2(棧底->棧頂) S1 (棧底->棧頂) 說明 5 5 空 數字，直接入棧 - 5 - S1為空，運算符直接入棧 ) 5 - ) 右括號直接入棧 4 5 4 - ) 數字直接入棧 × 5 4 - ) × S1棧頂是右括號，直接入棧 ) 5 4 - ) × ) 右括號直接入棧 3 5 4 3 - ) × ) 數字 + 5 4 3 - ) × ) + S1棧頂是右括號，直接入棧 2 5 4 3 2 - ) × ) + 數字 ( 5 4 3 2 + - ) × 左括號，彈出運算符直至遇到右括號 ( 5 4 3 2 + × - 同上 + 5 4 3 2 + × - + 優先級與-相同，入棧 1 5 4 3 2 + × 1 - + 數字 到達最左端 5 4 3 2 + × 1 + - 空 S1中剩余的運算符
因此結果為“- + 1 × + 2 3 4 5”。后綴表達式（后綴記法、逆波蘭式） 后綴表達式的計算機求值： 將中綴表達式轉換為后綴表達式：
掃描到的元素 S2(棧底->棧頂) S1 (棧底->棧頂) 說明 1 1 空 數字，直接入棧 + 1 + S1為空，運算符直接入棧 ( 1 + ( 左括號，直接入棧 ( 1 + ( ( 同上 2 1 2 + ( ( 數字 + 1 2 + ( ( + S1棧頂為左括號，運算符直接入棧 3 1 2 3 + ( ( + 數字 ) 1 2 3 + + ( 右括號，彈出運算符直至遇到左括號 × 1 2 3 + + ( × S1棧頂為左括號，運算符直接入棧 4 1 2 3 + 4 + ( × 數字 ) 1 2 3 + 4 × + 右括號，彈出運算符直至遇到左括號 - 1 2 3 + 4 × + - -與+優先級相同，因此彈出+，再壓入- 5 1 2 3 + 4 × + 5 - 數字 到達最右端 1 2 3 + 4 × + 5 - 空 S1中剩余的運算符
編寫Java程序將一個中綴表達式轉換為前綴表達式和后綴表達式，并計算表達式的值。其中的toPolishNotation()方法將中綴表達式轉換為前綴表達式（波蘭式）、toReversePolishNotation()方法則用于將中綴表達式轉換為后綴表達式（逆波蘭式）：
注：
[java] ?view plaincopy
package?qmk.simple_test;?? import?java.util.Scanner;?? import?java.util.Stack;?? public?class?Calculator?{?? ??????public?static?final?String?USAGE?=?"==?usage?==\n"?? ????????????+?"input?the?expressions,?and?then?the?program?"?? ????????????+?"will?calculate?them?and?show?the?result.\n"?? ????????????+?"input?'bye'?to?exit.\n";?? ?????? ??????public?static?void?main(String[]?args)?{?? ????????????System.out.println(USAGE);?? ????????????Scanner?scanner?=?new?Scanner(System.in);?? ????????????String?input?=?"";?? ????????????final?String?CLOSE_MARK?=?"bye";?? ????????????System.out.println("input?an?expression:");?? ????????????input?=?scanner.nextLine();?? ????????????while?(input.length()?!=?0?? ??????????????????&&?!CLOSE_MARK.equals((input)))?{?? ??????????????????System.out.print("Polish?Notation?(PN):");?? ??????????????????try?{?? ????????????????????????toPolishNotation(input);?? ??????????????????}?catch?(NumberFormatException?e)?{?? ????????????????????????System.out.println("\ninput?error,?not?a?number.");?? ??????????????????}?catch?(IllegalArgumentException?e)?{?? ????????????????????????System.out.println("\ninput?error:"?+?e.getMessage());?? ??????????????????}?catch?(Exception?e)?{?? ????????????????????????System.out.println("\ninput?error,?invalid?expression.");?? ??????????????????}?? ??????????????????System.out.print("Reverse?Polish?Notation?(RPN):");?? ??????????????????try?{?? ????????????????????????toReversePolishNotation(input);?? ??????????????????}?catch?(NumberFormatException?e)?{?? ????????????????????????System.out.println("\ninput?error,?not?a?number.");?? ??????????????????}?catch?(IllegalArgumentException?e)?{?? ????????????????????????System.out.println("\ninput?error:"?+?e.getMessage());?? ??????????????????}?catch?(Exception?e)?{?? ????????????????????????System.out.println("\ninput?error,?invalid?expression.");?? ??????????????????}?? ??????????????????System.out.println("input?a?new?expression:");?? ??????????????????input?=?scanner.nextLine();?? ????????????}?? ????????????System.out.println("program?exits");?? ??????}?? ?????? ??????private?static?void?toPolishNotation(String?input)?? ??????????????????throws?IllegalArgumentException,?NumberFormatException?{?? ????????????int?len?=?input.length();?? ????????????char?c,?tempChar;?? ????????????Stack<Character>?s1?=?new?Stack<Character>();?? ????????????Stack<Double>?s2?=?new?Stack<Double>();?? ????????????Stack<Object>?expression?=?new?Stack<Object>();?? ????????????double?number;?? ????????????int?lastIndex?=?-1;?? ????????????for?(int?i=len-1;?i>=0;?--i)?{?? ??????????????????c?=?input.charAt(i);?? ??????????????????if?(Character.isDigit(c))?{?? ????????????????????????lastIndex?=?readDoubleReverse(input,?i);?? ????????????????????????number?=?Double.parseDouble(input.substring(lastIndex,?i+1));?? ????????????????????????s2.push(number);?? ????????????????????????i?=?lastIndex;?? ????????????????????????if?((int)?number?==?number)?? ??????????????????????????????expression.push((int)?number);?? ????????????????????????else?? ??????????????????????????????expression.push(number);?? ??????????????????}?else?if?(isOperator(c))?{?? ????????????????????????while?(!s1.isEmpty()?? ????????????????????????????????????&&?s1.peek()?!=?')'?? ????????????????????????????????????&&?priorityCompare(c,?s1.peek())?<?0)?{?? ??????????????????????????????expression.push(s1.peek());?? ??????????????????????????????s2.push(calc(s2.pop(),?s2.pop(),?s1.pop()));?? ????????????????????????}?? ????????????????????????s1.push(c);?? ??????????????????}?else?if?(c?==?')')?{?? ????????????????????????s1.push(c);?? ??????????????????}?else?if?(c?==?'(')?{?? ????????????????????????while?((tempChar=s1.pop())?!=?')')?{?? ??????????????????????????????expression.push(tempChar);?? ??????????????????????????????s2.push(calc(s2.pop(),?s2.pop(),?tempChar));?? ??????????????????????????????if?(s1.isEmpty())?{?? ????????????????????????????????????throw?new?IllegalArgumentException(?? ??????????????????????????????????????????"bracket?dosen't?match,?missing?right?bracket?')'.");?? ??????????????????????????????}?? ????????????????????????}?? ??????????????????}?else?if?(c?==?'?')?{?? ???????????????????????? ??????????????????}?else?{?? ????????????????????????throw?new?IllegalArgumentException(?? ????????????????????????????????????"wrong?character?'"?+?c?+?"'");?? ??????????????????}?? ????????????}?? ????????????while?(!s1.isEmpty())?{?? ??????????????????tempChar?=?s1.pop();?? ??????????????????expression.push(tempChar);?? ??????????????????s2.push(calc(s2.pop(),?s2.pop(),?tempChar));?? ????????????}?? ????????????while?(!expression.isEmpty())?{?? ??????????????????System.out.print(expression.pop()?+?"?");?? ????????????}?? ????????????double?result?=?s2.pop();?? ????????????if?(!s2.isEmpty())?? ??????????????????throw?new?IllegalArgumentException("input?is?a?wrong?expression.");?? ????????????System.out.println();?? ????????????if?((int)?result?==?result)?? ??????????????????System.out.println("the?result?is?"?+?(int)?result);?? ????????????else?? ??????????????????System.out.println("the?result?is?"?+?result);?? ??????}?? ?????? ??????private?static?void?toReversePolishNotation(String?input)?? ??????????????????throws?IllegalArgumentException,?NumberFormatException?{?? ????????????int?len?=?input.length();?? ????????????char?c,?tempChar;?? ????????????Stack<Character>?s1?=?new?Stack<Character>();?? ????????????Stack<Double>?s2?=?new?Stack<Double>();?? ????????????double?number;?? ????????????int?lastIndex?=?-1;?? ????????????for?(int?i=0;?i<len;?++i)?{?? ??????????????????c?=?input.charAt(i);?? ??????????????????if?(Character.isDigit(c)?||?c?==?'.')?{?? ????????????????????????lastIndex?=?readDouble(input,?i);?? ????????????????????????number?=?Double.parseDouble(input.substring(i,?lastIndex));?? ????????????????????????s2.push(number);?? ????????????????????????i?=?lastIndex?-?1;?? ????????????????????????if?((int)?number?==?number)?? ??????????????????????????????System.out.print((int)?number?+?"?");?? ????????????????????????else?? ??????????????????????????????System.out.print(number?+?"?");?? ??????????????????}?else?if?(isOperator(c))?{?? ????????????????????????while?(!s1.isEmpty()?? ????????????????????????????????????&&?s1.peek()?!=?'('?? ????????????????????????????????????&&?priorityCompare(c,?s1.peek())?<=?0)?{?? ??????????????????????????????System.out.print(s1.peek()?+?"?");?? ??????????????????????????????double?num1?=?s2.pop();?? ??????????????????????????????double?num2?=?s2.pop();?? ??????????????????????????????s2.push(calc(num2,?num1,?s1.pop()));?? ????????????????????????}?? ????????????????????????s1.push(c);?? ??????????????????}?else?if?(c?==?'(')?{?? ????????????????????????s1.push(c);?? ??????????????????}?else?if?(c?==?')')?{?? ????????????????????????while?((tempChar=s1.pop())?!=?'(')?{?? ??????????????????????????????System.out.print(tempChar?+?"?");?? ??????????????????????????????double?num1?=?s2.pop();?? ??????????????????????????????double?num2?=?s2.pop();?? ??????????????????????????????s2.push(calc(num2,?num1,?tempChar));?? ??????????????????????????????if?(s1.isEmpty())?{?? ????????????????????????????????????throw?new?IllegalArgumentException(?? ??????????????????????????????????????????"bracket?dosen't?match,?missing?left?bracket?'('.");?? ??????????????????????????????}?? ????????????????????????}?? ??????????????????}?else?if?(c?==?'?')?{?? ???????????????????????? ??????????????????}?else?{?? ????????????????????????throw?new?IllegalArgumentException(?? ????????????????????????????????????"wrong?character?'"?+?c?+?"'");?? ??????????????????}?? ????????????}?? ????????????while?(!s1.isEmpty())?{?? ??????????????????tempChar?=?s1.pop();?? ??????????????????System.out.print(tempChar?+?"?");?? ??????????????????double?num1?=?s2.pop();?? ??????????????????double?num2?=?s2.pop();?? ??????????????????s2.push(calc(num2,?num1,?tempChar));?? ????????????}?? ????????????double?result?=?s2.pop();?? ????????????if?(!s2.isEmpty())?? ??????????????????throw?new?IllegalArgumentException("input?is?a?wrong?expression.");?? ????????????System.out.println();?? ????????????if?((int)?result?==?result)?? ??????????????????System.out.println("the?result?is?"?+?(int)?result);?? ????????????else?? ??????????????????System.out.println("the?result?is?"?+?result);?? ??????}?? ?????? ??????private?static?double?calc(double?num1,?double?num2,?char?op)?? ??????????????????throws?IllegalArgumentException?{?? ????????????switch?(op)?{?? ????????????case?'+':?? ??????????????????return?num1?+?num2;?? ????????????case?'-':?? ??????????????????return?num1?-?num2;?? ????????????case?'*':?? ??????????????????return?num1?*?num2;?? ????????????case?'/':?? ??????????????????if?(num2?==?0)?throw?new?IllegalArgumentException("divisor?can't?be?0.");?? ??????????????????return?num1?/?num2;?? ????????????default:?? ??????????????????return?0;? ????????????}?? ??????}?? ?????? ??????private?static?int?priorityCompare(char?op1,?char?op2)?{?? ????????????switch?(op1)?{?? ????????????case?'+':?case?'-':?? ??????????????????return?(op2?==?'*'?||?op2?==?'/'???-1?:?0);?? ????????????case?'*':?case?'/':?? ??????????????????return?(op2?==?'+'?||?op2?==?'-'???1?:?0);?? ????????????}?? ????????????return?1;?? ??????}?? ?????? ??????private?static?int?readDoubleReverse(String?input,?int?start)?? ??????????????????throws?IllegalArgumentException?{?? ????????????int?dotIndex?=?-1;?? ????????????char?c;?? ????????????for?(int?i=start;?i>=0;?--i)?{?? ??????????????????c?=?input.charAt(i);?? ??????????????????if?(c?==?'.')?{?? ????????????????????????if?(dotIndex?!=?-1)?? ??????????????????????????????throw?new?IllegalArgumentException(?? ????????????????????????????????????"there?have?more?than?1?dots?in?the?number.");?? ????????????????????????else?? ??????????????????????????????dotIndex?=?i;?? ??????????????????}?else?if?(!Character.isDigit(c))?{?? ????????????????????????return?i?+?1;?? ??????????????????}?else?if?(i?==?0)?{?? ????????????????????????return?0;?? ??????????????????}?? ????????????}?? ????????????throw?new?IllegalArgumentException("not?a?number.");?? ??????}?? ???????? ?????? ??????private?static?int?readDouble(String?input,?int?start)?? ??????throws?IllegalArgumentException?{?? ????????????int?len?=?input.length();?? ????????????int?dotIndex?=?-1;?? ????????????char?c;?? ????????????for?(int?i=start;?i<len;?++i)?{?? ??????????????????c?=?input.charAt(i);?? ??????????????????if?(c?==?'.')?{?? ????????????????????????if?(dotIndex?!=?-1)?? ??????????????????????????????throw?new?IllegalArgumentException(?? ??????????????????????????????"there?have?more?than?1?dots?in?the?number.");?? ????????????????????????else?if?(i?==?len?-?1)?? ??????????????????????????????throw?new?IllegalArgumentException(?? ??????????????????????????????"not?a?number,?dot?can't?be?the?last?part?of?a?number.");?? ????????????????????????else?? ??????????????????????????????dotIndex?=?i;?? ??????????????????}?else?if?(!Character.isDigit(c))?{?? ????????????????????????if?(dotIndex?==?-1?||?i?-?dotIndex?>?1)?? ??????????????????????????????return?i;?? ????????????????????????else?? ??????????????????????????????throw?new?IllegalArgumentException(?? ??????????????????????????????"not?a?number,?dot?can't?be?the?last?part?of?a?number.");?? ??????????????????}?else?if?(i?==?len?-?1)?{?? ????????????????????????return?len;?? ??????????????????}?? ????????????}?? ?????????????? ????????????throw?new?IllegalArgumentException("not?a?number.");?? ??????}?? ?????? ??????private?static?boolean?isOperator(char?c)?{?? ????????????return?(c=='+'?||?c=='-'?||?c=='*'?||?c=='/');?? ??????}?? }?? ?
下面是程序運行結果（綠色為用戶輸入）：
== usage == input the expressions, and then the program will calculate them and show the result. input 'bye' to exit.
?
input an expression: 3.8+5.3 Polish Notation (PN):+ 3.8 5.3 the result is 9.1 Reverse Polish Notation (RPN):3.8 5.3 + the result is 9.1 input a new expression: 5*(9.1+3.2)/(1-5+4.88) Polish Notation (PN):/ * 5 + 9.1 3.2 + - 1 5 4.88 the result is 69.88636363636364 Reverse Polish Notation (RPN):5 9.1 3.2 + * 1 5 - 4.88 + / the result is 69.88636363636364 input a new expression: 1+((2+3)*4)-5 Polish Notation (PN):- + 1 * + 2 3 4 5 the result is 16 Reverse Polish Notation (RPN):1 2 3 + 4 * + 5 - the result is 16 input a new expression: bye program exits
總結
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