Given an infinite cardinal $\kappa$ and a $\kappa$-splitting $\kappa^+$-tree $T$, we topologize $T$ as follows: if $x\in T$, then sets of the form $\uparrow x \setminus \uparrow F$, for $F$ a set of immediate successors of $x$ with $ F <\kappa$, form a basis of neighbourhoods of $x$. We then ask whether $T$ is $\kappa^+$-compact with respect to this topology and characterize this property in purely order-theoretic terms. Such trees are necessarily $\kappa^+$-Aronszajn, so they may (consistently) not exist when $\kappa\ge\aleph_1$. In this talk, discuss how to construct such trees using Proxy Principles, introduced by Brodsky and Rinot. We will also mention a further consitency result on the non-existence of such trees together with the failure of the tree property at $\aleph_2$. This is joint work with Ari Meir Brodksy.