Delete umbraco log files across all folders

$ago = 7;
$now = Get-Date;
$limit = $now.AddDays($ago * -1)
$filename = "UmbracoTraceLog.txt.*"
$path = "D:\Websites\"

# Delete files older than the $limit.
Get-ChildItem -Path $path -Recurse -Force | Where-Object { !$_.PSIsContainer -and $_.LastwriteTime -lt $limit -and $_.name -like $filename } | Remove-Item -WhatIf

# Delete any empty directories left behind after deleting the old files.
Get-ChildItem -Path $path -Recurse -Force | Where-Object { $_.PSIsContainer -and (Get-ChildItem -Path $_.FullName -Recurse -Force | Where-Object { !$_.PSIsContainer }) -eq $null } | Remove-Item -WhatIf