Could anyone explain me what the kernel build tree is?
What files does it include?
How can I obtain it from a cross-compiled kernel source (e.g. 3.4.0)?
well the kernel build tree, is the kernel source itself
to get it an "install" it you should:
I think you can download it as well via apt like
sudo apt-get install linux-source (if you need the 3.4.0 make sure that your debian has it as package)
With the information that you want to cross-compile your kernel to load it in an external device, I will assume that you want to do it for an arm-based device http://www.arm.linux.org.uk/docs/kerncomp.php if it's not, you can easily adapt it to another plataform