Wilf tree的结构类似二叉树,每个节点的数字能从其父节点推导而得,是用于求得所有在给定整数范围内的有理数的一种方法。其结构如下图所示,根节点为1,根节点即为该树的第一层,若给定整数为N,则1-N层所有节点即为N内的所有有理数。
父节点与子节点的关系:
设父节点的分子为i,分母为j,则左子节点为(i/(i+j)),右节点为(i+(i+j)/j),若还不能理解请参考维基百科:https://en.wikipedia.org/wiki/Calkin%E2%80%93Wilf_tree,有动画进行说明。
用Java进行实现,其实过程很简单,即去搜寻每个节点的两个子节点,将子节点都放入集合后再将父节点标记为访问过的节点,当父节点所在层数为给定整数N,停止往下计算。当所有节点都被标记为已访问,程序结束。
节点类:
/** * 有理数结点类 */ public class RationalNumber { private int num1; private int num2; private boolean flag = false; int layer; public void setNum1(int num1){ this.num1 = num1; } public void setNum2(int num2){ this.num2 = num2; } public void setFlag(boolean flag){ this.flag = flag; } public void setLayer(int layer){ this.layer = layer; } public int getNum1(){ return num1; } public int getNum2(){ return num2; } public boolean isFlag(){ return flag; } public int getLayer(){ return layer; } }
import java.util.ArrayList; import java.util.Scanner; /** * 利用Wilf tree列出所有在给定整数内的有理数 */ public class FindRationalNumber { public static void main(String[] args){ System.out.println("Please input an integer:"); Scanner scanner = new Scanner(System.in);//给定整数 int stopNumber = scanner.nextInt(); ArrayList<RationalNumber> list = new ArrayList();//存储所有符合要求的有理数 if(stopNumber>0){//当给定整数大于0,将1加入list并且开始搜寻符合要求的其他有理数 RationalNumber rationalNumber1 = new RationalNumber(); rationalNumber1.setNum1(1); rationalNumber1.setNum2(1); rationalNumber1.setLayer(1); list.add(rationalNumber1); for(int i =0; i < Math.pow(2,stopNumber-1);i++){ list = find(list.get(i),list,stopNumber); } } System.out.println("Rational Numbers:"); System.out.print("0"+" ");//单独将0输出,也可将其化为RationalNumber类加入list for(RationalNumber r : list){//输出结果 System.out.print(r.getNum1()+"/"+r.getNum2()+" "); } } /** * 计算节点rationlNumber的两个子节点并且存入list * @param rationalNumber * @param list * @param stopNumber * @return */ public static ArrayList<RationalNumber> find(RationalNumber rationalNumber, ArrayList<RationalNumber> list, int stopNumber){ if(rationalNumber.isFlag()|| rationalNumber.getLayer()>=stopNumber){ return list; } RationalNumber nextRationalNumber = new RationalNumber(); int temple = rationalNumber.getNum1()+ rationalNumber.getNum2(); nextRationalNumber.setNum1(rationalNumber.getNum1()); nextRationalNumber.setNum2(temple); nextRationalNumber.setLayer(rationalNumber.getLayer()+1); list.add(nextRationalNumber);//加入左子节点 RationalNumber nextRationalNumber2 = new RationalNumber(); nextRationalNumber2.setNum2(rationalNumber.getNum2()); nextRationalNumber2.setNum1(temple); nextRationalNumber2.setLayer(rationalNumber.getLayer()+1); list.add(nextRationalNumber2);//加入右子节点 rationalNumber.setFlag(true);//标记rationalNumber的子节点已计算过 return list; } }
输出结果: