php help with custom page navigation and subpages

I am trying to buid different navigation on different pages. I have used the below code and it works fine. However, I am unclear how to incorporate subpages.

For instance, i would like to write in php….

if the page being viewed is a subpage of birthday, then include links.php, but if it is a subpage of gifts then include links-home.php.

Can someone tell me how to include the above into php code?

Thank you….

if (is_page()) {

        // we're looking at a static page.  Which one?

        if (is_page('birthday')) {

             // our about page.
             include 'links.php'; }

        elseif (is_page('gifts')) {
             include 'links-home.php'; }

else {
        // catch-all for everything else (archives, searches, 404s, etc)

        include 'links-all.php';

} // That's all, folks!