  Suppose that $(x_s)_{s\in S}$ is a normalized family in a Banach space
indexed by the dyadic tree $S$. Using Stern's combinatorial theorem we extend
important results from sequences in Banach spaces to tree-families. More
precisely, assuming that for any infinite chain $\beta$ of $S$ the sequence
$(x_s)_{s\in\beta}$ is weakly null, we prove that there exists a subtree $T$ of
$S$ such that for any infinite chain $\beta$ of $T$ the sequence
$(x_s)_{s\in\beta}$ is nearly (resp., convexly) unconditional. In the case
where $(f_s)_{s\in S}$ is a family of continuous functions, under some
additional assumptions, we prove the existence of a subtree $T$ of $S$ such
that for any infinite chain $\beta$ of $T$, the sequence $(f_s)_{s\in\beta}$ is
unconditional. Finally, in the more general setting where for any chain
$\beta$, $(x_s)_{s\in\beta}$ is a Schauder basic sequence, we obtain a
dichotomy result concerning the semi-boundedly completeness of the sequences
$(x_s)_{s\in\beta}$.

Functional Analysis