  For a given graph, the unlabeled subgraphs $G-v$ are called the cards of $G$
and the deck of $G$ is the multiset $\{G-v: v \in V(G)\}$. Wendy Myrvold [Ars
Combinatoria, 1989] showed that a non-connected graph and a connected graph
both on $n$ vertices have at most $\lfloor \frac{n}{2} \rfloor +1$ cards in
common and she found (infinite) families of trees and non-connected forests for
which this upper bound is tight. Bowler, Brown, and Fenner [Journal of Graph
Theory, 2010] conjectured that this bound is tight for $n \geq 44$. In this
article, we prove this conjecture for sufficiently large $n$. The main result
is that a tree $T$ and a unicyclic graph $G$ on $n$ vertices have at most
$\lfloor \frac{n}{2} \rfloor+1$ common cards. Combined with Myrvold's work this
shows that it can be determined whether a graph on $n$ vertices is a tree from
any $\lfloor \frac{n}{2}\rfloor+2$ of its cards.
  Based on this theorem, it follows that any forest and non-forest also have at
most $\lfloor \frac{n}{2} \rfloor +1$ common cards. Moreover, we have
classified all except finitely many pairs for which this bound is strict.
Furthermore, the main ideas of the proof for trees are used to show that the
girth of a graph on $n$ vertices can be determined based on any $\frac{2n}{3}
+1$ of its cards. Lastly, we show that any $\frac{5n}{6} +2$ cards determine
whether a graph is bipartite.
