binary decision tree hyperplane parallel to axis? I'm reading the random forest paper.


  binary decision trees use a single feature at each nonterminal node. A test point is assigned to the left or right branch by its value of that feature. Geometrically this corresponds to assigning the point to one side of the hyperplane that is parallel to one axis of the feature space.


Can someone explain why the hyperplane is parallel to the axis? 

Say the feature is represented by the $x$ axis, and that the decision is $x &lt; 5$ vs $x &gt;5$. To me it seems like the hyperplane should be vertical to the $x$ axis so that $x = 4$ is on one side and $x=6$ is on the other. What's wrong with my understanding? Thanks.
Your understanding is fine. 

The hyperplane is $x=5$. 

In $3$ dimension, it is a plane that is parallel to the the $y$-$z$ plane, which can be written as $x=0$. It is parallel to the $y$-axis and $z$-axis.

You are right that it is perpendicular to the $x$-axis. The $x$-axis is the normal direction for the plane $x=5$.