For any ordinal $\lambda$ of uncountable cofinality, a $\lambda$-tree is a tree $T$ of height $\lambda$ such that $|T_{\alpha }|<\mathrm{cf}\left(\lambda \right)$ for each $\alpha <\lambda$, where $T_{\alpha }=\{x\in T:\mathrm{ht}\left(x\right)=\alpha \}$. In this note we get a Pressing Down Lemma for $\lambda$-trees and discuss some of its applications. We show that if $\eta$ is an uncountable ordinal and $T$ is a Hausdorff tree of height $\eta$ such that $|T_{\alpha }|\le \omega$ for each $\alpha <\eta$, then the tree $T$ is collectionwise Hausdorff if and only if for each antichain $C\subset T$ and for each limit ordinal $\alpha \le \eta$ with $\mathrm{cf}\left(\alpha \right)>\omega$, $\left\{\mathrm{ht}\right(c):c\in C\}\cap \alpha$ is not stationary in $\alpha$. In the last part of this note, we investigate some...