Cloning a JS TreeModel tree

I need to clone a tree I made using TreeModel.js. What I exactly need to do is duplicating it, make changes to it and check if the number of nodes decreased. If it did, revert to the original tree. Here's a small example of what I do so far to duplicate it, which is not correct:

var tree = new TreeModel();
var root = tree.parse({
    id: 0,
    name: "Root",
    children: [{id: 1, name: "1", children: []},{id: 2, name: "2", children: []}]
});

console.log(root)
var dup = tree.parse(root)
console.log(dup)

Here's a Fiddle. You'll see the difference between the trees by looking at the console:

Node {config: Object, model: Object, children: Array[2], isRoot: function, hasChildren: function…}
Node {config: Object, model: Node, children: Array[2], isRoot: function, hasChildren: function…}

Is there any way to properly clone such a structure? I looked for cloning JS object but still, I can't find a way for cloning this object exactly (such as the prototypes of properties like the model...)

You can deep clone the model of the first tree and parse it again to get a second tree.

Taking on your example:

function deepCopy(obj) {
    // You can also use the jquery extend method here
    return JSON.parse(JSON.stringify(obj));
}

var dup = tree.parse(deepCopy(root.model));

Important: If you do not deep clone the model, and just parse it again, you'll end up with the same underlying model shared by both trees which will certainly cause inconsistencies.

I finally came to a solution that may help anyone with the same problem:

var tree = new TreeModel();
var root = tree.parse({
    id: 0,
    name: "Root",
    children: [{id: 1, name: "1", children: []},{id: 2, name: "2", children: []}]
});

console.log(root)
var dup = tree.parse(root.model)
console.log(dup)

The parse function takes a model as parameter and the model of root seems to work fine.

EDIT: this solution may bring inconsistencies since the 2 trees are based on the same model. JNS's solution is more appropriate.

Why not trying a jQuery deep copy?