  We show that we can construct simultaneously all the stable trees as a nested
family. More precisely, if $1 < a < a' \leq 2$ we prove that hidden inside any
a-stable we can find a version of an a'-stable tree rescaled by an independent
Mittag-Leffler type distribution. This tree can be explicitly constructed by a
pruning procedure of the underlying stable tree or by a modification of the
fragmentation associated with it. Our proofs are based on a recursive
construction due to Marchal which is proved to converge almost surely towards a
stable tree.
