In your particular case what you want is Tree /f. You have a comment asking how to strip out the part at the front talking about the volume, serial number, and drive letter. That is possible filtering the output before you send it to file.
$Path = "C:\temp"
Tree $Path /F | Select-Object -Skip 2 | Set-Content C:\temp\output.tkt
Tree’s output in the above example is a System.Array which we can manipulate. Select-Object -Skip 2 will remove the first 2 lines containing that data. Also, If Keith Hill was around he would also recommend the PowerShell Community Extensions(PSCX) that contain the cmdlet Show-Tree. Download from here if you are curious. Lots of powerful stuff there.