Recursive function to find children count in javascript

It's fairly simple without any library or custom code.

//Or whatever way you want to grab a reference to the DOM node
var p1 = document.getElementById('P1');

p1.getElementsByTagName('*').length

Maybe I misunderstood the question. Is this what you want?:

p1.childNodes.length

Here is how you can do this using recursive Depth-first search:

function dfs(current, result) {
  var children = current.getChildren(),
      forEach = Array.prototype.forEach;
  result = result || {};
  result[current.id] = result[current.id] || 0;
  forEach.call(children, function (c) {
    result[current.id] += 1;
    dfs(c, result);
  });
  return result;
}
var result = dfs(document.getElementById('p1'));

console.log(result);

The script uses pure JavaScript.

Sample HTML:

<div id="p1">
  <div id="p2">
    <div id="p5"></div>
    <div id="p6"></div>
  </div>
  <div id="p3">
    <div id="p7"></div>
    <div id="p8"></div>
    <div id="p9"></div>
    <div id="p10">
        <div id="p11"></div>
    </div>
  </div>
  <div id="p4"></div>
</div>

result contains:

p1: 3
p2: 2
p3: 4
p4: 0
p5: 0
p6: 0
p7: 0
p8: 0
p9: 0
p10: 1
p11: 0

Note that you must have id of each element, otherwise you'll get results like:

undefined: 15
p1: 5
...

Here is an example in jsfiddle.

Here's a little recursive function, off the top of my head, though...

function recChildCount(node, level)
{
    level = (level || 0)++;
    var tree = {},
    i = 0;
    tree['level' + level] = {count: node.childNodes.length,
                             children: []};
    if (node.hasChildNodes() === false)
    {//dead end
        return tree;
    }
    for (i; i < node.childNodes.length; i++)
    {
        tree['level' + level].children.push(recChildCount(node.childNodes[i],level));
    }
    return tree;
}

This should return an object along the lines of:

{level1: {count: 123,
          level2: { count: 12,
                    children: [level3: { count: 16,
                                        children:[{level4:{count: 0,
                                                          children:[]},
                                                  {level4: {count: 0,
                                                            children:[]}]
                                        }
                               ]
                   }
           }
 }

Or something like that, you could add a delete tree.children; when there are no children, of course.
Again, this is just off the top of my head, so this code might need some fine-tuning, but I hope it's enough to help you on your way

Use jquery plugin

You can use .length with just a descendant selector, like this:

If you have to use .children(), then it's like this:

var count = $("#yourId").children().length;

You want only first level, so use

$('#yourId > *').children().length;