findItem(id, items = null) {
  if (!items) {
    items = this.items;
  }

  return items.reduce((acc, item) => {
    if (acc) {
      return acc;
    }

    if (item.id === id) {
      return item;
    }

    if (item.children) {
      return this.findItem(id, item.children);
    }

    return acc;
  }, null);
}

<v-app>
  <v-treeview :items="items">
    <template slot="append" slot-scope="{ item }">
      <v-btn @click="addChild(item);">Add child</v-btn>
    </template>
  </v-treeview>
</v-app>

addChild(item) {
  if (!item.children) {
    this.$set(item, "children", []);
  }

  const name = `${item.name} (${item.children.length})`;
  const id = this.nextId++;
  item.children.push({
    id,
    name
  });
}