In Unity you have full access to .net's File IO functions, which you can use to write any kind of data you like (providing you're not targeting the webplayer).

You may need to decide on some text-based format in which to encode your tree node data, and then write a 'reader' and a 'writer' which converts your tree data to and from the text format.

You could also google "c# reading and writing text files" which turns up a load of tutorials, most of which are applicable to Unity.

If you need more help, you might need to supply more detailed information about the kind of data that you want to save.

To traverse a tree system and convert it to a file-savable string, you'd want to use a recursive function call, where each call to convert a node also adds that nodes children to the string - something like this pseudocode, which uses a JSON-like format:

(untested - for illustrative purposes only)

public string Serialize() {
    StringBuilder s = new StringBuilder();
    SerializeNode(rootNode,s);
    return s.ToString();
}

public void SerializeNode(Node node, StringBuilder s)
{
    s.Append(node.name);
    s.Append(":[");
    bool firstChild = true;
    foreach (Node child in node.children) {
        if (!firstChild) {
            s.Append(",");
        }
        s.Append(SerializeNode(child,s);
        firstChild = false;
    }
    s.Append("]");
}