How to automatically set terminal title to directory name without path

Why do you want to do it without modifying your PS1 variable? That is the correct way to do it.

If you run the following, you'll get what you want:

export PS1="\[\e]0;\W\a\]${debian_chroot:+($debian_chroot)}\[\033[01;32m\]\u@\h\[\033[00m\]:\[\033[01;34m\]\w\[\033[00m\]\$ "

This is just taking the default prompt:

export PS1="\[\e]0;\u@\h: \w\a\]${debian_chroot:+($debian_chroot)}\[\033[01;32m\]\u@\h\[\033[00m\]:\[\033[01;34m\]\w\[\033[00m\]\$ "

... and replacing \u@\h: \w with \W.

You can do the same thing with any command that echoes the right escape code:

echo -en "\e]0;${PWD##*/}\a"

... but then you'd be constantly fighting bash to stop it from doing it using PS1 (the best way).

Yes. Prepend: \[\e]0;\W\a\] to your PS1. This will not change your prompts appearance, but will automatically set your terminal tabs title to the basename of the current working directory.