Please help me, java recursion problem

[rathrathon]

ok so i got into high school AP CPS and am currently learning java on eimacs.com, but there program is not all too help full when learning, at least for me any ways
i was hoping someone could explain some code to me, what it does etc, and then help me with a problem

for first the explanation

public static String reverseString( String s )
{
  if ( s.length() == 0 )
    return "";
 
  String firstChar = s.substring( 0, 1 );
  String reverseRest = reverseString( s.substring( 2 ) );
 
  String result = reverseRest + firstChar;
 
  return result;
}
 
public static void main( String[] args ) 
{
  String a = "The sky's the limit!";
  System.out.println( reverseString( a ) );
}

and then the problem i am trying to do is count vowels in a word, using recursion and not using for or while loops
my code is

public static int countVowels( String s )
{
int i = 0;
int total = 0;
 
String vowel = s.substring( i,i+1 );
 
if (s.equals("") )
    return 0;
    
if ( vowel == "a" ||  vowel == "A" ||  vowel == "e" ||  vowel == "E" ||  vowel == "i" ||  vowel == "I" ||  vowel == "o" ||  vowel == "O" ||  vowel == "u" ||  vowel == "U" )
 total++;
 i++;

 if ( i  <= s.length () )
 {
 i++;
  countVowels ( s.substring ( i, i + 1 ) );
 }
  else 
 return total ;
}

which always gives me a missing returns statement, or when i play with the code a stack overflow

any help please, i am so far behind in my class, and am so lost...

THANK YOU IN ADVANCE,
RATHRATHON

[Fou-Lu]

Recursion always works the same. Keep going until you find a satisfied result, otherwise, try again. With the first example, the check is on the length of the provided string, once it hits 0 there is no more recursion. Examples are hard to do though, so you need to follow a stack trace. With just the word 'The', you would end up with:

firstChar1 = 'T'
SUB
firstChar2 = 'e';
SUB
NULL

So the end result would be 'eT'. The answer appears wrong since the recursion only works from the third character on. That should be fetching from substring(1).
You never need to work with an increment when dealing with recursion. This is because the parameter should be modified to match only the remaining options. So with a vowel counter you would need

SUB COUNTVOWELS(STRING input)
    SET iTotal TO 0
    IF input.LENGTH > 0 THEN
        SET sEvaluate TO input.SUBSTRING(0, 1)
        IF sEvaluate.ISVOWEL() THEN
            SET iTotal TO 1
        ENDIF
    ENDIF
    SET iTotal TO iTotal + (CALL COUNTVOWELS(input.SUBSTRING(1)))
    RETURN iTotal
END SUB

ISVOWEL can actually be done from either the Character class, or by comparing. This is wrong though: if ( vowel == "a" || vowel == "A". Strings cannot be compared this way. If you stick to string, you need to use vowel.equals("a") or vowel.equalsIgnoreCase("a").

Edit:
Sorry, the above is pseudo code. Please implement it and come back if it gives you any trouble.