Remarks on the construction of $K_{\sigma}$ sets associated to trees not satisfying a separation condition

K_{\sigma} sets involving sticky maps \sigma have been used in the theory of differentiation of integrals to probabilistically construct Kakeya-type sets that imply certain types of directional maximal operators are unbounded on L^{p}(\mathbb{R}^{2}) for all 1 \leq p < \infty . We indicate limits to this approach by showing that, given \varepsilon > 0 and a natural number N , there exists a tree \mathcal{T}_{N, \varepsilon} of finite height that is lacunary of order N but such that, for every sticky map \sigma\colon \mathcal{B}^{h(\mathcal{T}_{N, \varepsilon})}\rightarrow \mathcal{T}_{N, \varepsilon} , one has |K_{\sigma}\cap ((1,2) \times \mathbb{R})| \geq 1 - \varepsilon .