4 problems

Let ${\widetilde X}_n(\alpha)$ be the normalized sum of powers of subtree sizes for a conditioned Galton--Watson tree. At the critical line, take a nonzero real number $t$ and the…

Let $\xi$ be an offspring distribution with $\operatorname{\mathbb E}\xi=1$, and let $\mathcal T_n$ be a conditioned Galton–Watson tree with offspring distribution $\xi$. For any f…

Noninteger-moment conjecture. The assumption that $r$ is an integer is unnecessary in Theorem.

Let ${\mathcal T}_n$ be the conditioned Galton–Watson tree with $n$ nodes, and let a node be called good when it satisfies the preceding goodness condition depending on constants…