  We study the behavior of Haar coefficients in Besov and Triebel-Lizorkin
spaces on $\mathbb{R}$, for a parameter range in which the Haar system is not
an unconditional basis. First, we obtain a range of parameters, extending up to
smoothness $s<1$, in which the spaces $F^s_{p,q}$ and $B^s_{p,q}$ are
characterized in terms of doubly oversampled Haar coefficients (Haar frames).
Secondly, in the case that $1/p<s<1$ and $f\in B^s_{p,q}$, we actually prove
that the usual Haar coefficient norm, $\|\{2^j\langle f,
h_{j,\mu}\rangle\}_{j,\mu}\|_{b^s_{p,q}}$ remains equivalent to
$\|f\|_{B^s_{p,q}}$, i.e., the classical Besov space is a closed subset of its
dyadic counterpart. At the endpoint case $s=1$ and $q=\infty$, we show that
such an expression gives an equivalent norm for the Sobolev space
$W^{1}_p(\mathbb{R})$, $1<p<\infty$, which is related to a classical result by
Bo\v{c}karev. Finally, in several endpoint cases we clarify the relation
between dyadic and standard Besov and Triebel-Lizorkin spaces.
