  We investigate some aspects of the problem of the estimation of birth
distributions (BD) in multi-type Galton-Watson (MGW) trees with unobserved
types. More precisely, we consider a 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-types 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 succeed 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 $\nu$.
