jsTree Open a branch

Your code for open branch is correct.

For example. Source of tree:

    <div id="treeTask">
       <ul>
          <li id="node_37"><a href="#">TEST1</a>
              <ul>
                  <li id="node_38"><a href="#">TEST2</a></li>
                  <li id="node_39"><a href="#">TEST3</a></li>
              </ul>
          </li>
      </ul>
   </div>

Open node:

$("#treeTask").jstree("open_node", $("#node_38"));

Here is function that can open a specific node and all its parents.

function expandNode(nodeID) {
    // Expand all nodes up to the root (the id of the root returns as '#')
    while (nodeID != '#') {
        // Open this node
        $("#jstree").jstree("open_node", nodeID)
        // Get the jstree object for this node
        var thisNode = $("#jstree").jstree("get_node", nodeID);
        // Get the id of the parent of this node
        nodeID = $("#jstree").jstree("get_parent", thisNode);
    }
}

You could use the binding

$("#tree").bind("open_node.jstree", function (event, data) { 
  if((data.inst._get_parent(data.rslt.obj)).length) { 
    data.inst._get_parent(data.rslt.obj).open_node(this, false); 
  } 
}); 

Try this code to open node till nth Level

$("#myTree").jstree({options}).bind('loaded.jstree', function (e, data) {
    /** 
     * Open nodes on load (until x'th level) 
     */
    var depth = 3;
    data.inst.get_container().find('li').each(function (i) {
        if (data.inst.get_path($(this)).length <= depth) {
            data.inst.open_node($(this));
        }
    });
});