Bipartite Turán Numbers of Trees and Star Forests

The bipartite Tur\'an number of a graph $H$, denoted $\text{ex}(m, n; H)$, is the maximum number of edges in any $H$-free bipartite graph $G = (A, B; E)$ with parts of size $|A| = m$ and $|B| = n$. We study this problem for two families. For a tree $T = T(r, s)$ with parts $R$ and $S$ of sizes $|R| = r \le s = |S|$, we prove $$(r - 1) n \;\le\; \text{ex}(m, n; T(r, s)) \;\le\; (r - 1) n + O(m)$$ for $n$ sufficiently large compared to $m$, $r$, and $s$, determining the leading-order term exactly (with the star case $r=1$ solved with an exact formula). For a star forest $F = \bigcup_{i=1}^k S_{d_i}$ with $d_1 \ge \cdots \ge d_k$, we determine the exact value $\text{ex}(m, n; F) = (k - 1) n + (d_k - 1)(m - k + 1)$ for $n$ sufficiently large, and characterize the unique extremal graph.