JavaScript assigning named function to a variable

When you instantiate a function with a name — a name after the function keyword, that is — then that name is bound inside the function as a reference to itself. Outside the function, the name is not visible unless the function is created in a function declaration statement. In your example, that's not the case, so the local variable "walk_the_DOM" is bound to the function.

The key advantage here is that the function so created will continue to work (assuming a copy of the reference to it is saved somewhere) even if "walk_the_DOM" changes its value somehow. Why? Because the name "walk" is bound inside the function, and will remain bound regardless of what happens to "walk_the_DOM".

So there are two parts to this statement:

var walk_the_DOM = function walk(node, func) {
    func(node);
    node = node.firstChild;
    while (node) {
        walk(node, func);
        node = node.nextSibling;
    }
};

The first part is

var walkt_the_DOM ... 

That's a plain var declaration, and it creates a variable that's not really special in any way. Its value is detrermined by the initialization expression:

... = function walk(node, func) {
        func(node);
        node = node.firstChild;
        while (node) {
            walk(node, func);
            node = node.nextSibling;
        }
    };

which happens to be a function instantiation.

What would happen if the function were declared without a name? Like this:

var walk_the_DOM = function(node, func) {
    func(node);
    node = node.firstChild;
    while (node) {
        walk_the_DOM(node, func); // NOTE CHANGE HERE
        node = node.nextSibling;
    }
};

That would work, unless for some reason "walk_the_DOM" changes:

var thatFunction = walk_the_DOM;

walk_the_DOM = "hello world";

thatFunction(whatever, someFunc); // WILL NOT WORK

We've saved a reference to the function, but because the function in this version expects to be able to "find itself" via the external (to the function) variable "walk_the_DOM", it fails, because "walk_the_DOM" no longer refers to the function. That's the advantage of giving the function its own name.

Here is a classic article on the topic. Note that that article points out some implementation bugs, but it's somewhat dated and I think modern browsers do better.