Magento – Category tree from specific parent category

if you are in the category view page you can do the following.

$mainCategory = Mage::registry('current_category');
foreach ($mainCategory->getChildrenCategories() as $category){
    $collection = $category->getProductCollection()->addAttributeToSelect('image');//or thumbnail or small_image
    Mage::getSingleton('catalog/product_status')->addVisibleFilterToCollection($collection);
    Mage::getSingleton('catalog/product_visibility')->addVisibleInCatalogFilterToCollection($collection); 
    $collection->addAttributeToSort('position');//or 'name' or any other attribute
    $product = $collection->setPage(1, 1)->getFirstItem();
    if ($product->getId()){//if the category has at least a valid product
       $imageUrl = Mage::>helper('catalog/image')->init($product, 'image')->resize(60);
       //do something with $imageUrl
    }
}

One reason for it going slow is the number of loads you are doing in a loop. Also you are loading the initial category when you do not need to.

I have taken the liberty of refactoring your function. Hope you don't mind :)

function getSubsName($product, $parentId)
{
    $content = "";

    // get the product categories
    $categories = $product->getCategoryIds();

    // get all children of the parent $parentId
    $children = Mage::getModel('catalog/category')->getCategories($parentId, 0, false, true);
    foreach ($children as $category) {

        // if the child is in the product categories,
        // include it
        if (in_array($category->getId(), $categories)) {
            $content .= '<a href="'.$category->getUrl().'">'.$category->getName().'</a>';
        }
    }

    return $content;
}