  A Boolean network (BN) is called observable if any initial state can be
uniquely determined from the output sequence. In the existing literature on
observability of BNs, there is almost no research on the relationship between
the number of observation nodes and the observability of BNs, which is an
important and practical issue. In this paper, we mainly focus on three types of
BNs with $n$ nodes (i.e., $K$-AND-OR-BNs, $K$-XOR-BNs, and $K$-NC-BNs, where
$K$ is the number of input nodes for each node and NC means nested canalyzing)
and study the upper and lower bounds of the number of observation nodes for
these BNs. First, we develop a novel technique using information entropy to
derive a general lower bound of the number of observation nodes, and conclude
that the number of observation nodes cannot be smaller than
$\left[(1-K)+\frac{2^{K}-1}{2^{K}}\log_{2}(2^{K}-1)\right]n$ to ensure that any
$K$-AND-OR-BN is observable, and similarly, some lower bound is also obtained
for $K$-NC-BNs. Then for any type of BN, we also develop two new techniques to
infer the general lower bounds, using counting identical states at time 1 and
counting the number of fixed points, respectively. On the other hand, we derive
nontrivial upper bounds of the number of observation nodes by combinatorial
analysis of several types of BNs. Specifically, we indicate that
$\left(\frac{2^{K}-K-1}{2^{K}-1}\right)n,~1$, and $\lceil \frac{n}{K}\rceil$
are the best case upper bounds for $K$-AND-OR-BNs, $K$-XOR-BNs, and $K$-NC-BN,
respectively.
