<?php    
 
$category_model = Mage::getModel('catalog/category'); //get category model
$_category = $category_model->load($categoryid); //$categoryid for which the child categories to be found        
$all_child_categories = $category_model->getResource()->getAllChildren($_category); //array consisting of all child categories id
 
?>

http://blog.decryptweb.com/child-categories-magento/

One approach to get all child categories of a magento category.

Get child categories in Magento

magento

PHP