Assigning div ids to pages and their children

I’ve been trying to figure out a way to assign particular header div ids to particular pages. I want my div id to not only be assigned to a particular page, but also to all of its children. In order to assign the div id to a page, I’ve used the following code:

<?php if (is_home() ) { ?>
    <div id="header">
<?php } else { ?>
    <div id="<?php echo $post->post_name; ?>">

The way it works, then, is that I have a div id with the same name as the page (page name = example, div id entry in CSS = #example). This has been working extremely well.

My question is, does anyone have a suggestion for how to set it up similarly, but have all of the page’s children recall the div id of their parents? (page’s parent = example, therefore page’s id recalled = #example) I’ve been trying to do this through using page_parent, but I just simply can’t figure it out!

Thanks so much!