In earlier works, we have developed a Monte Carlo method to compute the first eigenvalue of linear operators, which is based on the simulation of exit times. In this paper, we show how to use a branching method to handle in a better way the simulation of large exit times. We show furthermore that this new method provides naturally an estimation of the first eigenfunction of the adjoint operator. Numerical examples are given on the Laplace operator and on homogeneous neutron transport operators.