3 problems

Let ${\mathrm{w}}=(w_k,k \ge 0)$ be a weight sequence with $w_0>0$ and with $w_k>0$ for some $k \ge 2$. For $n \ge 0$ with $Z_n({\mathrm{w}})>0$, let ${\mathcal{T}}_n$ be a simply…

Let ${\mathcal T}_n$ be the conditioned simply generated tree, and let $H({\mathcal T}_n)$ denote its height. Assume that the condensation parameter satisfies $\nu=1$ and that the…

Let ${\mathcal T}_n$ be the conditioned simply generated tree, let $\widehat{{\mathcal T}}$ be its limiting size-biased tree, and let $l_k(T)$ denote the number of vertices at dist…