In this paper we study detection and reconstruction of planted structures in
Erd\H{o}s-R\'enyi random graphs. Motivated by a problem of communication
security, we focus on planted structures that consist in a tree graph. For
planted line graphs, we establish the following phase diagram. In a low density
region where the average degree λ of the initial graph is below some
critical value λc=1, detection and reconstruction go from impossible
to easy as the line length K crosses some critical value f(λ)ln(n),
where n is the number of nodes in the graph. In the high density region
λ>λc, detection goes from impossible to easy as K goes from
o(n) to ω(n), and reconstruction remains impossible so
long as K=o(n). For D-ary trees of varying depth h and 2≤D≤O(1),
we identify a low-density region λ<λD, such that the following
holds. There is a threshold h∗=g(D)ln(ln(n)) with the following properties.
Detection goes from feasible to impossible as h crosses h∗. We also show
that only partial reconstruction is feasible at best for h≥h∗. We
conjecture a similar picture to hold for D-ary trees as for lines in the
high-density region λ>λD, but confirm only the following part of
this picture: Detection is easy for D-ary trees of size ω(n),
while at best only partial reconstruction is feasible for D-ary trees of any
size o(n). These results are in contrast with the corresponding picture for
detection and reconstruction of {\em low rank} planted structures, such as
dense subgraphs and block communities: We observe a discrepancy between
detection and reconstruction, the latter being impossible for a wide range of
parameters where detection is easy. This property does not hold for previously
studied low rank planted structures