Where do (random) trees grow leaves?

We study a model of random binary trees grown “by the leaves" in the style of Luczak and Winkler [LW04]. If τ n is a uniform plane binary tree of size n , Luczak and Winkler, and later explicitly Caraceni and Stauffer, constructed a measure ν τ n such that the tree obtained by adding a cherry on a leaf sampled according to ν τ n is still uniformly distributed on the set of all plane binary trees with size n + 1 . It turns out that the measure ν τ n , which we call the leaf-growth measure, is noticeably different from the uniform measure on the leaves of the tree τ n . In fact we prove that, as n → ∞ , with high probability it is almost entirely supported by a subset of only n 3 ( 2 - 3 ) + o ( 1 ) ≈ n 0 . 8038 . . . leaves . In the continuous setting, we construct the scaling limit of uniform binary trees equipped with this measure, which is the Brownian Continuum Random Tree equipped with a probability measure supported by a fractal set of dimension 6 ( 2 - 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.