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There are some positive results about narrow classes of trees, as well as computational results for small $n$ (the best I could find is positive for all $n \leq 35$). … One could, however, ask for unconditional results about largest $|D|$ achievable for all trees with a given $n$. An easy greedy algorithm yields $|D|$ of size $n / 3$ for any tree of size $n$. …

Consider two random tree models $T_1(n)$ and $T_2(n)$, chosen equiprobably among labelled and unlabelled trees on $n$ vertices respectively. …

Let $T_n$ be a random tree on $n$ labelled vertices chosen equiprobably among all $n^{n - 2}$ trees, and $I(T)$ be the number of distinct independent sets of a tree $T$. … Here are the values for a few small values of $n$ (obtained by generating all trees and counting their independent sets with dynamic programming): $\mathrm{E}I(T_1) = 2$ $\mathrm{E}I(T_2) = 3$ $\mathrm …

If we restrict $G$ to belong to a certain class, some groups may become non-representable: e.g., automorphism groups of trees are the ones obtainable from the trivial group using direct product and wreath …