Planar Turán number of quasi-double stars

Given a graph H, we call a graph $\mathit{\text{H-free}}$ if it does not contain H as a subgraph. The planar Turán number of a graph H, denoted by $e{x}_{\mathcal{P}}(n,H)$, is the maximum number of edges in a planar H-free graph on n vertices. A (h,k)-quasi-double star $W_{h,k}$, obtained from a path $P_3=v_1{v}_2{v}_3$ by adding h leaves and k leaves to the vertices $v_1$ and $v_3$, respectively, is a subclass of caterpillars. In this paper, we study $e{x}_{\mathcal{P}}(n,W_{h,k})$ for all $1\le h\le 2\le k\le 5$, and obtain some tight bounds $e{x}_{\mathcal{P}}(n,W_{h,k})\le \frac{3(h+k)}{h+k+2}n$ for $3\le h+k\le 5$ with equality holds if $(h+k+2)\mid n$, and $e{x}_{\mathcal{P}}(n,W_{1,5})\le \frac{5}{2}n$ with equality holds if $12\mid n$. Also we show that $\frac{9}{4}n\le e{x}_{\mathcal{P}}(n,W_{2,4})\le \frac{5}{2}n$ and $\frac{5}{2}n\le e{x}_{\mathcal{P}}(n,W_{2,5})\le \frac{17}{6}n$, respectively.