I simplified the code, and replaced tr/()/<>/ to make it more readable:

$_='a<b<c<d><e>>f>g<h><<i>j>';
 
$re = qr{
          <                # anchor at first paren as  wanted
          (                # paren group 1 
            (?:
              (?> [^<>]+)  # Non-parens without backtracking
            |
              <
              (?1)         # Recurse to start of paren group 1
              >
            )*
          )
      }x;
 
/$re/;
print $1;
[download]

perl /tmp/tst2.pl
b<c<d><e>>f
[download]