  We consider the message complexity of verifying whether a given subgraph of
the communication network forms a tree with specific properties both in the
KT-$\rho$ (nodes know their $\rho$-hop neighborhood, including node IDs) and
the KT-$0$ (nodes do not have this knowledge) models. We develop a rather
general framework that helps in establishing tight lower bounds for various
tree verification problems. We also consider two different verification
requirements: namely that every node detects in the case the input is
incorrect, as well as the requirement that at least one node detects. The
results are stronger than previous ones in the sense that we assume that each
node knows the number $n$ of nodes in the graph (in some cases) or an $\alpha$
approximation of $n$ (in other cases). For spanning tree verification, we show
that the message complexity inherently depends on the quality of the given
approximation of $n$: We show a tight lower bound of $\Omega(n^2)$ for the case
$\alpha \ge \sqrt{2}$ and a much better upper bound (i.e., $O(n \log n)$) when
nodes are given a tighter approximation. On the other hand, our framework also
yields an $\Omega(n^2)$ lower bound on the message complexity of verifying a
minimum spanning tree (MST), which reveals a polynomial separation between ST
verification and MST verification. This result holds for randomized algorithms
with perfect knowledge of the network size, and even when just one node detects
illegal inputs, thus improving over the work of Kor, Korman, and Peleg (2013).
For verifying a $d$-approximate BFS tree, we show that the same lower bound
holds even if nodes know $n$ exactly, however, the lower bound is sensitive to
$d$, which is the stretch parameter.
