The construction of a spanning tree is a fundamental task in distributed systems which allows to resolve other tasks (i.e., routing, mutual exclusion, network reset). In this paper, we are interested in the problem of constructing a \emph{Breadth First Search} (BFS) tree. \emph{Stabilization} is a versatile technique which ensures that the system recover a correct behaviour from an arbitrary global state resulting from transient faults. A \emph{silent} algorithm always reaches a terminal global state in a finite time. We present a first silent stabilizing algorithm to resolve a problem in which each node requests a permission (delivered by a subset of network nodes) in order to perform a defined computation. Using this first algorithm, we present a silent stabilizing algorithm constructing a BFS tree working in $O(D^2)$ rounds ($D$ is the diameter of the network) under a distributed daemon without any fairness assumptions. The complexity in terms of steps is $O(mn^4)$ where $m$ and $n$ are the number of edges and nodes of the network, respectively, so it is polynomial with respect to $n$. To our knowledge, since in general the diameter of a network is much smaller than the number of nodes, this algorithm gets the best compromise of the literature between the complexities in terms of rounds and in terms of steps.