When you're using jquery make sure you're using mouseenter and mouseleave instead of mouseover and mouseout in these cases. mouseover and mouseout fire even when entering a child element. If you have code to hide things, you wouldn't want it to run in a menu at that point, you want it to run when you leave the <li> completely and consider a child as still being inside...that's what mouseenter and mouseleave do :)
There's also a shortcut for this called .hover() which uses the mouseenter and mouseleave events like this:
$(selector).hover(mouseenterFunc, mouseleaveFunc);
Here's an example:
$("li").hover(function() {
$(this).children("ul").slideDown();
}, function() {
$(this).children().slideUp();
});
You can see the above code in a quick demo against your posted tree here