Penalization of Galton-Watson processes

We apply the penalization technique introduced by Roynette, Vallois, Yor for Brownian motion to Galton-Watson processes with a penalizing function of the form $P (x)s^x$ where P is a polynomial of degree p and s $\in$ [0, 1]. We prove that the limiting martingales obtained by this method are most of the time classical ones, except in the super-critical case for s = 1 (or s $\rightarrow$ 1) where we obtain new martingales. If we make a change of probability measure with this martingale, we obtain a multi-type Galton-Watson tree with p distinguished infinite spines.