I'd use a hash to keep the structure:

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

<DATA>; # Skip header.

my %tree;
while (<DATA>) {
    chomp;
    my ($child, $parent, $relation) = split /,\t/;
    $relation =~ s/,$//;
    $tree{$child}{parent} = [$parent, $relation];
}

for my $node (keys %tree) {
    my @path      = $node;
    my @relations = ($tree{$node}{parent}[1]);
    my $parent    = $node;

    while ($parent = $tree{$parent}{parent}[0]) {
        push @path, $parent;
        push @relations, $tree{$parent}{parent}[1];
    }

    print "$node:\t";
    print join '<-', @path;
    print " = ", join '.', reverse grep defined, @relations;
    print "\n";
}

__DATA__
child,  Parent, relation
M10,    Q,    P,
M143,    M10,    P,
M406,    M143,    PL,
M407,    M143,    PL,
M420,    M143,    E,
M421,    M143,    E,
[download]

Output:

M143:   M143<-M10<-Q = P.P
M420:   M420<-M143<-M10<-Q = P.P.E
M407:   M407<-M143<-M10<-Q = P.P.PL
M10:    M10<-Q = P
M421:   M421<-M143<-M10<-Q = P.P.E
M406:   M406<-M143<-M10<-Q = P.P.PL
[download]