  We investigate some aspects of the problem of the estimation of birth
distributions (BD) in multi-type Galton-Watson trees (MGW) with unobserved
types. More precisely, we consider two-type MGW called spinal-structured trees.
This kind of tree is characterized by a spine of special individuals whose BD
$\nu$ is different from the other individuals in the tree (called normal whose
BD is denoted $\mu$). In this work, we show that even in such a very structured
two-type population, our ability to distinguish the two types and estimate
$\mu$ and $\nu$ is constrained by a trade-off between the growth-rate of the
population and the similarity of $\mu$ and $\nu$. Indeed, if the growth-rate is
too large, large deviations events are likely to be observed in the sampling of
the normal individuals preventing us to distinguish them from special ones.
Roughly speaking, our approach succeeds if $r<\mathfrak{D}(\mu,\nu)$ where $r$
is the exponential growth-rate of the population and $\mathfrak{D}$ is a
divergence measuring the dissimilarity between $\mu$ and $\nu$.
