Assuming that the GetProjectTypeData action takes a parameter called id then this should work:
$(".common").prev('h3').click(function() {
var $tree = $(this).next('.common');
var currentId = $tree.attr("id");
$tree.dynatree({
checkbox: true,
icon: false,
initAjax: {
type: "POST",
url: '/Home/GetProjectTypeData',
data: { id : currentId }
}
});
});