  We consider the transmission of a state from the root of a tree towards its
leaves, assuming that each transmission occurs through a noisy channel. The
states at the leaves are observed, while at deeper nodes we can compute the
likelihood of each state given the observation. In this sense, information
flows from child nodes towards the parent node. Here we find an upper bound of
this children-to-parent information flow. To do so, first we introduce a new
measure of information, the memory vector, whose norm quantifies whether all
states have the same likelihood. Then we find conditions such that the norm of
the memory vector at the parent node can be linearly bounded by the sum of
norms at the child nodes. We also describe the reconstruction problem of
estimating the ancestral state at the root given the observation at the leaves.
We infer sufficient conditions under which the original state at the root
cannot be confidently reconstructed using the observed leaves, assuming that
the number of levels from the root to the leaves is large.
