  We consider the problem of topology recognition in wireless (radio) networks
modeled as undirected graphs. Topology recognition is a fundamental task in
which every node of the network has to output a map of the underlying graph
i.e., an isomorphic copy of it, and situate itself in this map. In wireless
networks, nodes communicate in synchronous rounds. In each round a node can
either transmit a message to all its neighbors, or stay silent and listen. At
the receiving end, a node $v$ hears a message from a neighbor $w$ in a given
round, if $v$ listens in this round, and if $w$ is its only neighbor that
transmits in this round. Nodes have labels which are (not necessarily
different) binary strings. The length of a labeling scheme is the largest
length of a label. We concentrate on wireless networks modeled by trees, and we
investigate two problems.
  \begin{itemize} \item What is the shortest labeling scheme that permits
topology recognition in all wireless tree networks of diameter $D$ and maximum
degree $\Delta$?
  \item What is the fastest topology recognition algorithm working for all
wireless tree networks of diameter $D$ and maximum degree $\Delta$, using such
a short labeling scheme? \end{itemize}
  We are interested in deterministic topology recognition algorithms. For the
first problem, we show that the minimum length of a labeling scheme allowing
topology recognition in all trees of maximum degree $\Delta \geq 3$ is
$\Theta(\log\log \Delta)$. For such short schemes, used by an algorithm working
for the class of trees of diameter $D\geq 4$ and maximum degree $\Delta \geq
3$, we show almost matching bounds on the time of topology recognition: an
upper bound $O(D\Delta)$, and a lower bound $\Omega(D\Delta^{\epsilon})$, for
any constant $\epsilon<1$.
