# Article

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Keywords:
tree; $D$-space; $\lambda$-tree; property $\gamma$; collectionwise Hausdorff
Summary:
For any ordinal $\lambda$ of uncountable cofinality, a $\lambda$-tree is a tree $T$ of height $\lambda$ such that $|T_{\alpha }|<{\rm cf}(\lambda )$ for each $\alpha <\lambda$, where $T_{\alpha }=\{x\in T\colon {\rm ht}(x)=\alpha \}$. 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 $|T_{\alpha }|\leq \omega$ 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 $\alpha \leq \eta$ with ${\rm cf}(\alpha )>\omega$, $\{{\rm ht}(c)\colon c\in C\} \cap \alpha$ is not stationary in $\alpha$. In the last part of this note, we investigate some properties of $\kappa$-trees, $\kappa$-Suslin trees and almost $\kappa$-Suslin trees, where $\kappa$ is an uncountable regular cardinal.
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