  Recent theoretical work has shown that the pre-main-sequence (PMS) evolution
of stars is much more complex than previously envisioned. Instead of the
traditional steady, one-dimensional solution, accretion may be episodic and not
necessarily symmetrical, thereby affecting the energy deposited inside the star
and its interior structure. Given this new framework, we want to understand
what controls the evolution of accreting stars. We use the MESA stellar
evolution code with various sets of conditions. In particular, we account for
the (unknown) efficiency of accretion in burying gravitational energy into the
protostar through a parameter, $\xi$, and we vary the amount of deuterium
present. We confirm the findings of previous works that the evolution changes
significantly with the amount of energy that is lost during accretion. We find
that deuterium burning also regulates the PMS evolution. In the low-entropy
accretion scenario, the evolutionary tracks in the H-R diagram are
significantly different from the classical tracks and are sensitive to the
deuterium content. A comparison of theoretical evolutionary tracks and
observations allows us to exclude some cold accretion models ($\xi\sim 0$) with
low deuterium abundances. We confirm that the luminosity spread seen in
clusters can be explained by models with a somewhat inefficient injection of
accretion heat. The resulting evolutionary tracks then become sensitive to the
accretion heat efficiency, initial core entropy, and deuterium content. In this
context, we predict that clusters with a higher D/H ratio should have less
scatter in luminosity than clusters with a smaller D/H. Future work on this
issue should include radiation-hydrodynamic simulations to determine the
efficiency of accretion heating and further observations to investigate the
deuterium content in star-forming regions. (abbrev.)
