Create a rest route with response categories in hiearachical structure

wp_hierachy_categegories.php

<?php

/**
 * Class GK_Category_Walker
 */
class GK_Category_Walker extends Walker_Category{
  public function start_el( &$output, $category, $depth = 0, $args = array(), $id = 0 ) {
    /** This filter is documented in wp-includes/category-template.php */
    $cat_name = apply_filters(
      'list_cats',
      esc_attr( $category->name ),
      $category
    );
    $depth_prefix = '';
    $li_data = '';

    for( $i = 0; $i < $depth; $i++ ){
      $depth_prefix .= '-';
    }
    $li_data .= ' depth_prefix="' . $depth_prefix . '" ';
    $li_data .= ' depth="' . $depth . '" ';
    foreach( $category as $key => $value ) {
      if( $key === 'parent' ){
        if( $value != 0 ){
          $parent_cat = get_category($value);
          $li_data .= '' . $key . '_slug="' . $parent_cat->slug . '" ';
        }
      }
      $li_data .= '' . $key . '="' . $value . '" ';
    }

    // Don't generate an element if the category name is empty.
    if ( ! $cat_name ) {
      return;
    }

    $link = '<a href="' . esc_url( get_term_link( $category ) ) . '" ';
    if ( $args['use_desc_for_title'] && ! empty( $category->description ) ) {
      /**
       * Filters the category description for display.
       *
       * @since 1.2.0
       *
       * @param string $description Category description.
       * @param object $category    Category object.
       */
      $link .= 'title="' . esc_attr( strip_tags( apply_filters( 'category_description', $category->description, $category ) ) ) . '"';
    }

    $link .= '>';
    $link .= $cat_name . '</a>';

    if ( ! empty( $args['feed_image'] ) || ! empty( $args['feed'] ) ) {
      $link .= ' ';

      if ( empty( $args['feed_image'] ) ) {
        $link .= '(';
      }

      $link .= '<a href="' . esc_url( get_term_feed_link( $category->term_id, $category->taxonomy, $args['feed_type'] ) ) . '"';

      if ( empty( $args['feed'] ) ) {
        $alt = ' alt="' . sprintf(__( 'Feed for all posts filed under %s' ), $cat_name ) . '"';
      } else {
        $alt = ' alt="' . $args['feed'] . '"';
        $name = $args['feed'];
        $link .= empty( $args['title'] ) ? '' : $args['title'];
      }

      $link .= '>';

      if ( empty( $args['feed_image'] ) ) {
        $link .= $name;
      } else {
        $link .= "<img src='" . $args['feed_image'] . "'$alt" . ' />';
      }
      $link .= '</a>';

      if ( empty( $args['feed_image'] ) ) {
        $link .= ')';
      }
    }

    if ( ! empty( $args['show_count'] ) ) {
      $link .= ' (' . number_format_i18n( $category->count ) . ')';
    }
    if ( 'list' == $args['style'] ) {
      $output .= "\t<li ${li_data} ";
      $css_classes = array(
        'cat-item',
        'cat-item-' . $category->term_id,
      );

      if ( ! empty( $args['current_category'] ) ) {
        // 'current_category' can be an array, so we use `get_terms()`.
        $_current_terms = get_terms( $category->taxonomy, array(
          'include' => $args['current_category'],
          'hide_empty' => false,
        ) );

        foreach ( $_current_terms as $_current_term ) {
          if ( $category->term_id == $_current_term->term_id ) {
            $css_classes[] = 'current-cat';
          } elseif ( $category->term_id == $_current_term->parent ) {
            $css_classes[] = 'current-cat-parent';
          }
          while ( $_current_term->parent ) {
            if ( $category->term_id == $_current_term->parent ) {
              $css_classes[] =  'current-cat-ancestor';
              break;
            }
            $_current_term = get_term( $_current_term->parent, $category->taxonomy );
          }
        }
      }

      /**
       * Filters the list of CSS classes to include with each category in the list.
       *
       * @since 4.2.0
       *
       * @see wp_list_categories()
       *
       * @param array  $css_classes An array of CSS classes to be applied to each list item.
       * @param object $category    Category data object.
       * @param int    $depth       Depth of page, used for padding.
       * @param array  $args        An array of wp_list_categories() arguments.
       */
      $css_classes = implode( ' ', apply_filters( 'category_css_class', $css_classes, $category, $depth, $args ) );

      $output .=  ' class="' . $css_classes . '"';
      $output .= ">$link\n";
    } elseif ( isset( $args['separator'] ) ) {
      $output .= "\t$link" . $args['separator'] . "\n";
    } else {
      $output .= "\t$link<br />\n";
    }
  }
}

/**
 * Return WordPress categories in hierarchy array or object
 *
 * @param bool $to_array true - return array. false - return object(SimpleXMLElement)
 * @return array|mixed|object|SimpleXMLElement
 */
function gk_categories_in_hierarchy( $to_array = true ) {

  $xml = simplexml_load_string( wp_list_categories( array(
    'echo'    => 0,
    'walker'  => new GK_Category_Walker()
  ) ) );

  return $to_array ? json_decode(json_encode($xml), true) : $xml;
}

add_action( 'rest_api_init', function() {
  $namespace = 'gk/v1';
  $rest_base = 'site-data';

  register_rest_route( $namespace, $rest_base, array(
    'categories' => gk_categories_in_hierarchy()
  ));
});