Find the deepest left most node in a binary tree (and it's depth). A working demo can be found at: http://jsfiddle.net/YfLx6/5/

deepestNode.js

var Node = function(name, left, right) {
        this.name = name;
        this.left = left;
        this.right = right;
    },
    findDeepest = function(node, depth) {
        var left, right, depth = depth || 0;
        if(typeof node !== 'undefined') {
            left = findDeepest(node.left, depth+1);
            right = findDeepest(node.right, depth+1);
            if(typeof left !== 'undefined' && typeof right !== 'undefined') {
                if(left.depth >= right.depth) {
                    return left;
                } else {
                    return right;
                }
            } else if(typeof left !== 'undefined') {
                return left;
            } else if(typeof right !== 'undefined') {
                return right;
            } else {
                return {node: node, depth: depth};
            }
        }
    };

//Now to test it
var deepest = findDeepest(
        new Node('A',
               /*   \
              /      \
             /        \
            /        */
new Node('B'), new Node('C',
                      /*   \
                     /      \
                    /        \
                   /        */
      new Node('D'), new Node('E'))));

alert("Found '" + deepest.node.name + "' at depth of "+deepest.depth);
//returns "Found 'D' at depth of 2"