Gridlines in MulticolumnTreeView

How can I add gridlines to the MulticolumnTreeView?

You can partially achieve this by applying a border to your cell templates:

<DataTemplate x:Key="EnglishNameTemplate">
  <Border BorderBrush="AliceBlue" BorderThickness="0,0,1,1" Margin="-6,0,-6,0" Padding="6,0,6,0">
    <TextBlock Text="{Binding EnglishName}" FontFamily="Times New Roman" FontSize="12" />
  </Border>
</DataTemplate>

Note you will need to use a slightly different margin and padding on the first column to allow for the expander glyph.

However if your columns don't fill the entire width of the control then this will leave empty whitespace at the right hand end.  If you want the lines to extend all the way to the right then you need to template the MulticolumnTreeViewItem:

<Style x:Key="ics" TargetType="ms:MulticolumnTreeViewItem">
  <Setter Property="Template">
    <Setter.Value>
      <ControlTemplate TargetType="ms:MulticolumnTreeViewItem">
        <StackPanel>
          <Border BorderBrush="AliceBlue" BorderThickness="0,0,0,1">
            <GridViewRowPresenter x:Name="PART_Header" Content="{TemplateBinding Header}"
                                  Columns="{Binding WrappedColumns, RelativeSource={RelativeSource AncestorType={x:Type ms:MulticolumnTreeView}}}" />
          </Border>
          <ItemsPresenter Name="ItemsHost" />
        </StackPanel>
        <ControlTemplate.Triggers>
          <Trigger Property="IsExpanded" Value="False">
            <Setter TargetName="ItemsHost" Property="Visibility" Value="Collapsed"/>
          </Trigger>
          <MultiTrigger>
            <MultiTrigger.Conditions>
              <Condition Property="HasHeader" Value="False"/>
              <Condition Property="Width" Value="Auto"/>
            </MultiTrigger.Conditions>
            <Setter TargetName="PART_Header" Property="MinWidth" Value="75"/>
          </MultiTrigger>
          <MultiTrigger>
            <MultiTrigger.Conditions>
              <Condition Property="HasHeader" Value="False"/>
              <Condition Property="Height" Value="Auto"/>
            </MultiTrigger.Conditions>
            <Setter TargetName="PART_Header" Property="MinHeight" Value="19"/>
          </MultiTrigger>
          <Trigger Property="IsEnabled" Value="False">
            <Setter Property="Foreground" Value="{DynamicResource {x:Static SystemColors.GrayTextBrushKey}}"/>
          </Trigger>
        </ControlTemplate.Triggers>
      </ControlTemplate>
    </Setter.Value>
  </Setter>
</Style>

<ms:MulticolumnTreeView ItemContainerStyle="{StaticResource ics}" />

In this case you will not need the bottom part of the item border and can change the cell template border thicknesses to "0,0,1,0".