The following looks to do the shortest path variant:

#!/usr/bin/perl
use warnings;
use strict;

my %group = (    # Hash table/dictionary for all the  groups
    'P'        => 'I_1',
    'Pl'       => 'I_2',
    'P.P'      => 'I_3',
    'P.Pl'     => 'I_4',
    'Pl.P'     => 'I_5',
    'Pl.Pl'    => 'I_6',
    'P.P.P'    => 'I_7',
    'P.P.Pl'   => 'I_8',
    'P.Pl.P'   => 'I_9',
    'P.Pl.Pl'  => 'I_10',
    'Pl.P.P'   => 'I_11',
    'Pl.P.Pl'  => 'I_12',
    'Pl.Pl.P'  => 'I_13',
    'Pl.Pl.Pl' => 'I_14',
    'E'        => 'II_15',
    'P.E'      => 'II_16',
    'Pl.E'     => 'II_17',
    'P.P.E'    => 'II_18',
    'P.Pl.E'   => 'II_19',
    'Pl.P.E'   => 'II_20',
    'Pl.Pl.E'  => 'II_21',
    'E.P'      => 'III_22',
    'E.Pl'     => 'III_23',
    'P.E.P'    => 'III_24',
    'P.E.Pl'   => 'III_25',
    'Pl.E.P'   => 'III_26',
    'Pl.E.Pl'  => 'III_27',
    'E.P.P'    => 'III_28',
    'E.P.Pl'   => 'III_29',
    'E.Pl.P'   => 'III_30',
    'E.Pl.Pl'  => 'III_31',
    'E.E'      => 'IV_32',
    'P.E.E'    => 'IV_33',
    'Pl.E.E'   => 'IV_34',
    'E.P.E'    => 'IV_35',
    'E.Pl.E'   => 'IV_36',
    'E.E.P'    => 'IV_37',
    'E.E.Pl'   => 'IV_38',
    'E.E.E'    => 'IV_39',
);

<DATA>;    # Skip the headers (first row).

my %tree;
while (<DATA>) {
    # parse through the input data and fill in our tree data structure
    chomp;
    my ($child, $parent, $prob) = split /\t/;

    if ($child eq 'Q') {
        push @{$tree{$child}}, {parent => '', prob => $prob, dist => 0
+};
        next;
    }

    if ($parent eq 'Q') {
        push @{$tree{$child}}, {parent => $parent, prob => $prob, dist
+ => 1};
        next;
    }

    for my $opt (@{$tree{$parent}}) {
        my $dist = $opt->{dist} + 1;
        push @{$tree{$child}},
            {parent => $parent, prob => $prob, dist => $dist};
    }
}

for my $child (sort {length $a <=> length $b or $a cmp $b} keys %tree)
+ {
    my @bestPath = findBestPath($child, \%tree);
    my $probs = join '.', map {$_->{prob}} @bestPath;
    printf "%-5s ", "$child:";

    # Join the likelihood path. Then if group is found for a likelihoo
+d
    #from the group hash table then print it, else quit
    print join '<-', $child, grep {$_} map {$_->{parent}} @bestPath;
    print ", $probs";
    print ", $group{$probs}" if exists $group{$probs};
    print "\n";
}

sub findBestPath {
    my ($child, $tree) = @_;

    return $tree->{Q}[0] if $child eq 'Q';

    my @alts = sort {$a->{dist} <=> $b->{dist}} @{$tree->{$child}};
    return $alts[0], findBestPath($alts[0]{parent}, $tree);
}

__DATA__
child,    Parent,    likelihood
M7    Q    P
M54    M7    Pl
M213    M54    E
M206    M54    E
M194    M54    E
...
[download]

Prints (in part):

Q:    Q, E, II_15
M6:   M6<-Q, E.E, IV_32
M7:   M7<-Q, P.E, II_16
M10:  M10<-Q, E.E, IV_32
M13:  M13<-M7<-Q, E.P.E, IV_35
M17:  M17<-Q, P.E, II_16
M18:  M18<-Q, E .E
M22:  M22<-Q, E.E, IV_32
M23:  M23<-Q, E.E, IV_32
M28:  M28<-M6<-Q, P.E.E, IV_33
M33:  M33<-M28<-M6<-Q, E.P.E.E
[download]