[bes]

<div style="background: gray">div</div>
<script>
document.body.children[0].onmouseover = function () {
	if (!document.getElementById('di')) {
		this.insertAdjacentHTML('afterBegin', '<button id="di">click</button>');
	} 
}
document.body.children[0].onmouseout = function (e) {
	e = e || event;
	var rt = e.relatedTarget || e.toElement;
	while(rt && rt != this) {
		rt = rt.parentNode;
	}
	if (rt == this ) {
		return;
	} else if (document.getElementById('di')) {
		this.removeChild(document.getElementById('di'));
	}
}
</script>