  In the recursive lamination of the disk, one tries to add chords one after
another at random; a chord is kept and inserted if it does not intersect any of
the previously inserted ones. Curien and Le Gall [Ann. Probab. 39 (2011)
2224-2270] have proved that the set of chords converges to a limit
triangulation of the disk encoded by a continuous process $\mathscr{M}$. Based
on a new approach resembling ideas from the so-called contraction method in
function spaces, we prove that, when properly rescaled, the planar dual of the
discrete lamination converges almost surely in the Gromov-Hausdorff sense to a
limit real tree $\mathscr{T}$, which is encoded by $\mathscr{M}$. This confirms
a conjecture of Curien and Le Gall.
