  We study a model of random binary trees grown "by the leaves" in the style of
Luczak and Winkler. If $\tau_n$ is a uniform plane binary tree of size $n$,
Luczak and Winkler, and later explicitly Caraceni and Stauffer, constructed a
measure $\nu_{\tau_n}$ such that the tree obtained by adding a cherry on a leaf
sampled according to $\nu_{\tau_n}$ is still uniformly distributed on the set
of all plane binary trees with size $n+1$. It turns out that the measure
$\nu_{\tau_n}$, which we call the leaf-growth measure, is noticeably different
from the uniform measure on the leaves of the tree $\tau_n$. In fact, we prove
that, as $n \to \infty$, with high probability it is almost entirely supported
by a subset of only $n^{3 ( 2 - \sqrt{3})+o(1)} \approx n^{0.8038...}$ leaves.
In the continuous setting, we construct the scaling limit of this measure,
which is a probability measure on the Brownian Continuum Random Tree supported
by a fractal set of dimension $ 6 (2 - \sqrt{3})$. We also compute the full
(discrete) multifractal spectrum. This work is a first step towards
understanding the diffusion limit of the discrete leaf-growth procedure.
