question

Hi, at first I recommend to change the XAML an then include additional logic in ViewModel. Try demo.

XAML:

 <Window x:Class="Window016"
         xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"
         xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"
         xmlns:d="http://schemas.microsoft.com/expression/blend/2008"
         xmlns:mc="http://schemas.openxmlformats.org/markup-compatibility/2006"
         xmlns:local="clr-namespace:WpfApp1.WpfApp016"
         xmlns:mod="clr-namespace:WpfApp1.WpfApp016"
         mc:Ignorable="d"
         Title="Window016" Height="450" Width="800">
   <Window.Resources>
      <local:ViewModel x:Key="vm"/>
    <Style x:Key="HighlightingTextStyle" TargetType="Label">
       <Setter Property="FontStyle" Value="Italic"/>
     </Style>
   </Window.Resources>
   <Grid DataContext="{StaticResource vm}">
     <TreeView ItemsSource="{Binding TreeViewDataCollection}">
       <TreeView.Resources>
         <HierarchicalDataTemplate ItemsSource="{Binding HostGroupName}" DataType="{x:Type mod:RootNameList}">
           <StackPanel Orientation="Horizontal">
             <Label Style="{StaticResource HighlightingTextStyle}" Content="{Binding RootSubnet}"/>
           </StackPanel>
           </HierarchicalDataTemplate>
         <HierarchicalDataTemplate ItemsSource="{Binding Host}" DataType="{x:Type mod:HostGroup}">
           <StackPanel Orientation="Horizontal">
             <CheckBox IsChecked="{Binding IsParentChecked}" IsThreeState="True"/>
             <Label Style="{StaticResource HighlightingTextStyle}" Content="{Binding GroupName}" Tag="{Binding RelativeSource={RelativeSource AncestorType=Window}}"/>
           </StackPanel>
           </HierarchicalDataTemplate>
         <HierarchicalDataTemplate DataType="{x:Type mod:HostName}">
           <StackPanel Orientation="Horizontal">
             <CheckBox IsChecked="{Binding IsChildChecked}"/>
             <Label Style="{StaticResource HighlightingTextStyle}" Content="{Binding Host}" Tag="{Binding RelativeSource={RelativeSource AncestorType=Window}}"/>
           </StackPanel>
         </HierarchicalDataTemplate>
       </TreeView.Resources>
     </TreeView>
   </Grid>
 </Window>

Code:

 Imports System.ComponentModel
 Imports System.Runtime.CompilerServices
    
 Namespace WpfApp016
    
   Public Class ViewModel
    
     Public Sub New()
       For i = 1 To 10
         Dim rnl As New RootNameList With {.RootSubnet = $"RootSubnet{i}"}
         For k = 1 To 10
           Dim hg As New HostGroup With {.GroupName = $"GroupName {k + i * 10}"}
           rnl.HostGroupName.Add(hg)
           For l = 1 To 10
             Dim h As New HostName() With {.Host = $"Host {l}"}
             AddHandler h.PropertyChanged, AddressOf hg.h_PropertyChanged
             hg.Host.Add(h)
           Next
         Next
         col.Add(rnl)
       Next
       cvs.Source = col
     End Sub
    
     Private cvs As New CollectionViewSource
     Private col As New AsyncObservableCollection(Of RootNameList)
     Public ReadOnly Property TreeViewDataCollection As ICollectionView
       Get
         Return cvs.View
       End Get
     End Property
   End Class
    
   Public Class RootNameList
     Public Property RootSubnet As String
     Public Property HostGroupName As New List(Of HostGroup)
   End Class
    
   Public Class HostGroup
     Implements INotifyPropertyChanged
    
     Private _IsParentChecked As Boolean? = False
     Public Property IsParentChecked As Boolean?
       Get
         Return Me._IsParentChecked
       End Get
       Set(value As Boolean?)
         If Not value.HasValue Then value = False
         Me._IsParentChecked = value
         For Each h In Host
           h.IsChildChecked = value
         Next
         OnPropertyChanged()
       End Set
     End Property
     Public Property GroupName As String
     Public Property Host As New List(Of HostName)
    
     Friend Sub h_PropertyChanged(sender As Object, e As PropertyChangedEventArgs)
       Dim iFalse As Integer = 0
       Dim iTrue As Integer = 0
       For Each h In Host
         If h.IsChildChecked Then iTrue += 1 Else iFalse += 1
       Next
       If iTrue > 0 And iFalse = 0 Then Me._IsParentChecked = True
       If iTrue = 0 And iFalse > 0 Then Me._IsParentChecked = False
       If iTrue > 0 And iFalse > 0 Then Me._IsParentChecked = Nothing
       OnPropertyChanged(NameOf(IsParentChecked))
     End Sub
    
     Public Event PropertyChanged As PropertyChangedEventHandler Implements INotifyPropertyChanged.PropertyChanged
     Private Sub OnPropertyChanged(<CallerMemberName> Optional propName As String = "")
       RaiseEvent PropertyChanged(Me, New PropertyChangedEventArgs(propName))
     End Sub
   End Class
    
   Public Class HostName
     Implements INotifyPropertyChanged
    
     Private _IsChildChecked As Boolean? = False
     Public Property IsChildChecked As Boolean?
       Get
         Return Me._IsChildChecked
       End Get
       Set(value As Boolean?)
         Me._IsChildChecked = value
         OnPropertyChanged()
       End Set
     End Property
     Public Property Host As String
    
     Public Event PropertyChanged As PropertyChangedEventHandler Implements INotifyPropertyChanged.PropertyChanged
     Private Sub OnPropertyChanged(<CallerMemberName> Optional propName As String = "")
       RaiseEvent PropertyChanged(Me, New PropertyChangedEventArgs(propName))
     End Sub
   End Class
    
 End Namespace