You can combine the sparse checkout and therefore the shallow clone options.
The shallow clone shuts off/cuts off the history and the sparse checkout only pulls the files matching your patterns.
git init <repo>
git remote add origin <url>
git config core.sparsecheckout true
echo "finisht/*" >> .git/info/sparse-checkout
git pull --depth=1 origin master
Ensure that the minimum of git version of 1.9 is installed for this to work. Tested it myself only with 2.2.0 and 2.2.2. This way you will still be able to push, which isn't attainable with git archive.
For more information on frequently used commands visit git commands.