code: private void registerLocalChildBinding(Key<?> key, GinjectorBindings binding) {
      boundLocallyInChildren.put(key, binding);
      if (parent != null) {
        parent.registerLocalChildBinding(key, binding);
      }
    }
docs: Register the key in the "boundLocallyInChildren" set for this injector, and
  recursively register it with all of the ancestors.  The caller is
  responsible for ensuring that the binding being registered is actually
  local (i.e., not a ParentBinding).