  We study the maximal displacement and related population for a branching
Brownian motion in Euclidean space in terms of the principal eigenvalue of an
associated Schr\"odinger type operator. We first determine their growth rates
on the survival event. We then establish the upper deviation for the maximal
displacement under the possibility of extinction. Under the non-extinction
condition, we further discuss the decay rate of the upper deviation probability
and the population growth at the critical phase.
