I'd be inclined to use hashes instead of arrays. The code simplifies somewhat:

use strict;
use warnings;
use Data::Dump::Streamer;

my $s      = "f b, b c, a c, e b, d a, g a";
my %children = map {split ' ', $_} split ',', $s;
my %tree;

# Build the tree
push @{$tree{$children{$_}}}, $_ for keys %children;

# Find the root
my $root = (keys %children)[0];

while ($root) {
    last unless exists $children{$root};
    $root = $children{$root};
}

print "Root is $root\n";
Dump (\%tree);
[download]

Prints:

Root is c
$HASH1 = {
           a => [
                  'g',
                  'd'
                ],
           b => [
                  'e',
                  'f'
                ],
           c => [
                  'a',
                  'b'
                ]
         };
[download]