We study time and message complexity of the problem of building a BFS tree by a spontaneously awaken node in ad hoc network. Computation is in synchronous rounds, and messages are sent via point-to-point bi-directional links. Network topology is modeled by an undirected graph. Each node knows only its own id and the id's of its neighbors in the network and no pre-processing is allowed; therefore the solutions to the problem of spanning a BFS tree in this setting must be distributed. We deliver a deterministic distributed solution that trades time for messages, mainly, with time complexity $O(\diam\cdot\min(\diam,n/f(n)) \cdot \log \diam \cdot \log n)$ and with the number of point-to-point messages sent $O(n\cdot(\min(\diam,n/f(n)) + f(n)) \cdot \log \diam \cdot \log n)$, for any $n$-node network with diameter $\diam$ and for any monotonically non-decreasing sub-linear integer function $f$. Function $f$ in the above formulas come from the threshold value on node degrees used by our algorithms, in the sense that nodes with degree at most $f(n)$ are treated differently that the other nodes. This yields the first BFS-finding deterministic distributed algorithm in ad hoc networks working in time $o(n)$ and with $o(n^2)$ message complexity, for some suitable functions $f(n)=o(n/\log^2 n)$, provided $\diam=o(n/\log^4 n)$.