On the Distribution of Leaves in Rooted Subtrees of Recursive Trees

We study the structure of T⁽ᵏ⁾ₙ, the subtree rooted at k in a random recursive tree of order n, under the assumption that k is fixed and n → ∞. Employing generalized Polya urn models, exact and limiting distributions are derived for the size, the number of leaves and the number of internal nodes of T⁽ᵏ⁾ₙ. The exact distributions are given by intricate formulas involving Eulerian numbers, but a recursive argument based on the urn model suffices for establishing the first two moments of the above-mentioned random variables. Known results show that the limiting distribution of the size of T⁽ᵏ⁾ₙ, normalized by dividing by n is Beta(1, k - 1). A martingale central limit argument is used to show that the difference between the number of leaves and the number of internal nodes of T⁽ᵏ⁾ₙ, suitably normalized, converges to a mixture of normals with a Beta(1, k - 1) as the mixing density. The last result allows an easy determination of limiting distributions of suitably normalized versions of the number of leaves and the number of internal nodes of T⁽ᵏ⁾ₙ.