  We present a construction of a L\'evy continuum random tree (CRT) associated
with a super-critical continuous state branching process using the so-called
exploration process and a Girsanov's theorem. We also extend the pruning
procedure to this super-critical case. Let $\psi$ be a critical branching
mechanism. We set $\psi_\theta(\cdot)=\psi(\cdot+\theta)-\psi(\theta)$. Let
$\Theta=(\theta_\infty,+\infty)$ or $\Theta=[\theta_\infty,+\infty)$ be the set
of values of $\theta$ for which $\psi_\theta$ is a branching mechanism. The
pruning procedure allows to construct a decreasing L\'evy-CRT-valued Markov
process $(\ct_\theta,\theta\in\Theta)$, such that $\mathcal{T}_\theta$ has
branching mechanism $\psi_\theta$. It is sub-critical if $\theta>0$ and
super-critical if $\theta<0$. We then consider the explosion time $A$ of the
CRT: the smaller (negative) time $\theta$ for which $\mathcal{T}_\theta$ has
finite mass. We describe the law of $A$ as well as the distribution of the CRT
just after this explosion time. The CRT just after explosion can be seen as a
CRT conditioned not to be extinct which is pruned with an independent intensity
related to $A$. We also study the evolution of the CRT-valued process after the
explosion time. This extends results from Aldous and Pitman on Galton-Watson
trees. For the particular case of the quadratic branching mechanism, we show
that after explosion the total mass of the CRT behaves like the inverse of a
stable subordinator with index 1/2. This result is related to the size of the
tagged fragment for the fragmentation of Aldous' CRT.
