We study the asymptotic behaviour of the martingale ($ψ$ n (o)) n$\in$N associated with the Vertex Reinforced Jump Process (VRJP). We show that it is bounded in L p for every p > 1 on trees and uniformly integrable on Z d in all the transient phase of the VRJP. Moreover, when the VRJP is recurrent on trees, we have good estimates on the moments of $ψ$ n (o) and we can compute the exact decreasing rate $τ$ such that n --1 ln($ψ$ n (o)) $\sim$ --$τ$ almost surely where $τ$ is related to standard quantities for branching random walks. Besides, on trees, at the critical point, we show that n --1/3 ln($ψ$ n (o)) $\sim$ --$ρ$ c almost surely where $ρ$ c can be computed explicitely. Furthermore, at the critical point, we prove that the discrete process associated with the VRJP is a mixture of positive recurrent Markov chains. Our proofs use properties of the $β$-potential associated with the VRJP and techniques coming from the domain of branching random walks.