More specifically, I'm interested to know if there is a term that describes a tree node that meets two conditions: - It has >= one child - Any children that it has are all leaf nodes.
They are classified as nodes of height 1 by convention in the case of a rooted tree.
The height of a node $N$ is the number of edges on the longest downward simple path from $N$ to a leaf, according to Introduction to Algorithms by CLRS as well as the Wikipedia entry.
You might want to use penultimate nodes as suggested by Orlp. Please note some papers define a penultimate node slight differently on general graphs as a nonleaf node that is adjacent to a leaf node. For example, Patrolling Games.