How can I safely resize (shrink) a BTRFS partition?

According to this page in the btrfs wiki:

To shrink the filesystem by 4 GiB: btrfs filesystem resize -4g /mnt or btrfsctl -r -4g /mnt Set the FS size . To set the filesystem to a specific size, omit the leading + or - from the size.

The btrfs and btrfsctl commands in Ubuntu is provided by the package called btrfs-utils, which you can install in the Software Center. You run these commands in the Terminal (Ctrl+Alt+T). Replace 4g by however much you really want to shrink the partition, and /mnt by the mount point of the partition. For example, to shrink a btrfs partition mounted at /media/Alpha by 500 MiB, you would run:

btrfs filesystem resize -500m /media/Alpha

Or:

btrfsctl -r -500m /media/Alpha

Assuming the guidelines for resizing btrfs filesystems are the same as for other filesystems, you should avoid resizing a partition that is on the same physical disk as the running system. So, if necessary, boot from a live system to resize a partition on the drive that has your installed Ubuntu system. (This applies to resizing done in GParted, too.)

If some utilities or applications don't see the results of the resizing operation when it is performed, rebooting is a good idea.

See the btrfs and btrfsctl manual pages for more details about the use of these commands. And hopefully the btrfs wiki will be back up soon.

I am not personally very familiar with btrfs filesystems, but it seems strange to me that these resize commands take the mount point of the filesystem rather than the device name. If anyone can explain that, please comment or feel free to edit this answer to improve it.

According to the GParted features page, btrfs shrink is supported. If the option isn't available, installing the btrfs-tools package may fix it. If that doesn't work, it could be because Ubuntu includes an older version of GParted, in which case you could use the GParted Live CD:

http://gparted.sourceforge.net/livecd.php

(which is also how Ubuntu recommends using it anyway)