On the number of vertices with a given degree in a Galton-Watson tree

Let Y_k (ω) (k ≥ 0) be the number of vertices of a Galton-Watson tree ω that have k children, so that Z (ω) := ∑_k≥0 Y_k(ω) is the total progeny of w. In this paper, we will prove various statistical properties of Z and Y_k. We first show, under a mild condition, an asymptotic expansion of P(Z = n) as n → ∞, improving the theorem of Otter (1949). Next, we show that y_k(ω) := ∑ _j=0^k Y_j (ω) is the total progeny of a new Galton-Watson tree that is hidden in the original tree w. We then proceed to study the joint probability distribution of Z and {Y_k}_k, and show that, as n → ∞, {Y_k/n}_k is asymptotically Gaussian under the conditional distribution P(· Z = n).