  We consider a Brownian tree consisting of a collection of one-dimensional
Brownian paths started from the origin, whose genealogical structure is given
by the Continuum Random Tree (CRT). This Brownian tree may be generated from
the Brownian snake driven by a normalized Brownian excursion, and thus yields a
convenient representation of the so-called Integrated Super-Brownian Excursion
(ISE), which can be viewed as the uniform probability measure on the tree of
paths. We discuss different approaches that lead to the definition of the
Brownian tree conditioned to stay on the positive half-line. We also establish
a Verwaat-like theorem showing that this conditioned Brownian tree can be
obtained by re-rooting the unconditioned one at the vertex corresponding to the
minimal spatial position. In terms of ISE, this theorem yields the following
fact: Conditioning ISE to put no mass on $]-\infty,-\epsilon[$ and letting
$\epsilon$ go to 0 is equivalent to shifting the unconditioned ISE to the right
so that the left-most point of its support becomes the origin. We derive a
number of explicit estimates and formulas for our conditioned Brownian trees.
In particular, the probability that ISE puts no mass on $]-\infty,-\epsilon[$
is shown to behave like $2\epsilon^4/21$ when $\epsilon$ goes to 0. Finally,
for the conditioned Brownian tree with a fixed height $h$, we obtain a
decomposition involving a spine whose distribution is absolutely continuous
with respect to that of a nine-dimensional Bessel process on the time interval
$[0,h]$, and Poisson processes of subtrees originating from this spine.
