What does barium decompose to?

This depends entirely on which isotope of barium you're evaluating. 

Isotopes are variations in the neutron content of an atom; the most stable form of barium, Barium-130, has 56 protons and 74 neutrons. The number of protons will remain 56 in all isotopes, but the number of neutrons varies from 58 to 97. As you might expect, the further from having 74 neutrons the isotope is, in either direction, the shorter the half-life of the isotope tends to be. Most of the half-lives are measured in minutes or fractions of a second, whereas Barium-130 has a half-life tens of thousands of times longer than the age of the universe itself.

Most of the lighter isotopes decay into xenon and cesium, atoms one and two protons lighter than barium on the periodic table, and most of them decay through beta decay and electron capture. Beta-plus decay might be best described if you imagine the proton and being composed of a neutron plus a positively charged small particle (a positron); if the proton loses its positive charge, or absorbs a negative charge, it becomes a neutron. In beta-plus decay and electron capture, the proton loses its positive charge and becomes a neutron, dropping the atomic number by one. The heavier isotopes of barium also undergo beta decay, but they undergo beta-minus; in this case, the neutron can be thought of as a proton and an electron combined so that their charges cancel, and by losing an electron, the neutron becomes a proton, and the atomic number increases by one. The heavier isotopes thus tend to decay into a heavier element, lanthanum.