In a tree-like proof what is the term for the number of steps between any node and the root (the statement proved by the proof)? That is, if numb were this term, then one would have depth of tree = max(numb(leaf)), where the max is taken over all leaves in the tree.
The steps between the root node and a leaf node (or more precisely, a maximal linearly ordered subset of the set of nodes) is called a branch. The number of steps (minus 1) is called the length of the branch - this is the term you're looking for. The depth or height of a tree is the maximum length of it's branches, as you correctly conjectured. The length of a tree would be the number of nodes in it.