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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.

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    $\begingroup$ Maybe penultimate node? $\endgroup$ – orlp Jan 4 at 16:52
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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.

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