Linear Tur\'an Numbers of Uniform Hypertrees

A hypergraph is \emph{linear} if every pair of vertices is contained in at most one hyperedge. For a family $\mathcal{F}$ of $r$-uniform hypergraphs, the linear Tur'an number $ex^{\mathrm{lin}}_r(n,\mathcal{F})$ is the maximum number of hyperedges in an $n$-vertex $\mathcal{F}$-free linear $r$-uniform hypergraph. Extending the work of Gy'arf'as, Ruszink'o, and S'ark"ozy on $3$-uniform linear hypertrees, we study linear Tur'an numbers for higher uniformity.
We determine the linear Tur'an number of the $r$-uniform linear star $S_k^r$, proving [ex^{\mathrm{lin}}_r(n,S_k^r)\le \frac{n(k-1)}{r},] with equality exactly for $(k-1)$-regular linear $r$-uniform hypergraphs, whenever they exist. We also construct dense $T_k^r$-free hypergraphs showing that, under suitable divisibility and design-existence assumptions, [ex^{\mathrm{lin}}_r(n,T_k^r)\ge \frac{n(k-1)}{r}] for every linear $r$-uniform hypertree $T_k^r$ with $k$ hyperedges.
We then study all linear hypertrees with four hyperedges. For the broom $B_4^r$, we prove [ex^{\mathrm{lin}}_r(n,B_4^r)\le \frac{(r+1)n}{r},] and characterize the extremal hypergraphs as disjoint unions of Steiner systems $S(2,r,r^2)$, whenever such systems exist. For the crown $E_4^r$, we establish [ex^{\mathrm{lin}}_r(n,E_4^r)\le \frac{(2r-1)n}{r},] together with a lower-bound construction leaving only a constant-factor gap.
For the linear path $P_4^r$, we construct $P_4^r$-free hypergraphs with $(r+1)n/r$ hyperedges and conjecture this is the optimal general bound. We verify the conjecture for connected hypergraphs under suitable degree conditions. Finally, for $r=4$, we identify counterexamples to a key structural claim in a previous proof of Zhang and Wang, give a new proof that [ex^{\mathrm{lin}}_4(n,P_4^4)\le \frac{5n}{4},] and show that equality holds precisely for disjoint unions of Steiner systems $S(2,4,16)$.