I suspect the runtime error is coming from vertically merged cells rather than multiple instances of Tree Number. Can you confirm that your tables have no vertically merged cells?
To continue working on the same table, this change to Paul's code appears to work
Code:
Sub SplitTables()
Application.ScreenUpdating = False
Dim t As Long, c As Long
With ActiveDocument
On Error Resume Next
For t = .Tables.Count To 1 Step -1
RestartTable:
With .Tables(t).Range
For c = .Cells.Count To 1 Step -1
With .Cells(c)
If .RowIndex > 1 Then
If Split(.Range.Text, vbCr)(0) = "Tree Number" Then
.Range.InsertBreak (wdColumnBreak)
GoTo RestartTable
End If
End If
End With
Next
End With
Next
End With
Application.ScreenUpdating = True
End Sub