Does your brain hurt when you think of trees? Mine too... I don't ever want to write this again.

count_children.js

function countChildren (node) {
  function _countChildren (node) {
    if (node.children) {
      return node.children.map((child) => {
        return _countChildren(child);
      }).reduce((memo, next) => {
        return memo + next;
      }, 1);
    }
    return 1;
  }
  return _countChildren(node) - 1;
}