Drupal - How do I limit the allowed taxonomy terms basing on the content type?

According to my choice you should use hook_form_alter to bind taxonomy list based on article type.

function MY_MODULE_NAME_form_alter(&$form, &$form_state, $form_id) {
  if ($form_id == 'my_form_id') {
    $default_value = 0;
    $options = getArticles('Computer','Articles');
    $form['your_taxonomy_field_name'] = array(
      '#title' => t('Articlas'),
      '#type' => 'select',
      '#options' => $options,
      '#access' => 1,
      '#default_value' => $default_value,
      '#weight' => 1,
    );
  }
}
function getArticles($articleType,$vocabularyName) {
  //get the vocabulary
  $vocabulary = taxonomy_get_vocabularies();
  $vid = -1;
  foreach ($vocabulary as $item) {
    if ($item->name == $vocabularyName) {
      $vid = $item->vid;
    }
  }
  if ($vid != -1) {
    //get the parrent Term
    $query = new EntityFieldQuery;
    $result = $query
      ->entityCondition('entity_type', 'taxonomy_term')
      ->propertyCondition('name', $articleType)
      ->propertyCondition('vid', $vid)
      ->execute();
    $parrent_tid = array_shift($result['taxonomy_term'])->tid;
    //return children
    return taxonomy_get_children($parrent_tid);
  }
  return NULL;
}

The Term Reference Tree module might help. This module provides an expandable tree widget for the Taxonomy Term Reference field in Drupal 7.

Term Reference Tree:

This module provides an expandable tree widget for the Taxonomy Term Reference field in Drupal 7. This widget is intended to serve as a replacement for Drupal's core Taxonomy Term Reference widget, which is a flat list of radio buttons or checkboxes and not necessarily fit for medium to large taxonomy trees.

How to use this module

1) Enable the Term Reference Tree module.
2) Go to the Manage Fields tab of any fieldable entity (such as a content type, taxonomy term, or user.
3) Add a new field with a type of Term Reference.
4) In the widget select box, select Term reference tree.

5) Last step is important onc, as shown in the below image you have to give the taxonomy term id of which vocabulary you want to show for specific content type.

According to this answer on the main Stackoverflow site, a submodule of Content Taxonomy called "Content Taxonomy Options"would be perfect for this:

You can do this with Content Taxonomy's Content Taxonomy Options module. Add a new Content Taxonomy field, and use the Select widget. In the field settings, under the Advanced settings for hierarchical vocabularies field group, select the Parent Term you want to use for that field.