According to Wikipedia,
The height of a tree is the length of the path from the root to the deepest node in the tree. A (rooted) tree with only one
One advantage of using a node count rather than an edge count is that it distinguishes the empty case (zero nodes, and node level) from the minimal case (one node, and a node level of one). In some cases, an empty tree will not be meaningful, but in other cases an empty try will be perfectly legitimate.