In this article, we propose a least squares method for the estimation of the transition density in bifurcating Markov models. Unlike the kernel estimation, this method do not use the quotient which can be a source of errors. In order to study the rate of convergence for least squares estimators, we develop exponential inequalities for empirical process of bifurcating Markov chain under bracketing assumption. Unlike the classical processes, we observe that for bifurcating Markov chains, the complexity parameter depends on the ergodicity rate and as consequence, we have that the convergence rate of our estimator is a function of the ergodicity rate. We conclude with a numerical study to validate our theoretical results.