Branching random walk with a critical branching part

We consider the branching tree T(n) of the first (n + 1) generations of a critical branching process, conditioned on survival till time beta n for some fixed beta > 0 or on extinction occurring at time k(n), with k(n)/n --> beta beta. We attach to each vertex upsilon of this tree a random variable X(upsilon) and define S(upsilon) = Sigma(w is an element of pi) X(w), where pi(0, upsilon) is the unique path in the family tree from its root to upsilon. Finally M(n) is the maximal displacement of the branching random walk S(.), that is M(n) = max{S(upsilon): upsilon is an element of T(n)}. We show that if the X(upsilon), upsilon is an element of T(n), are i.i.d. with mean 0, then under some further moment condition n(-1/2)M(n) converges in distribution. In particular {n(-1/2)M(n)}(n greater than or equal to 1) is a tight family. This is closely related to recent results about Aldous' continuum tree and Le Gall's Brownian snake.