1. Height of one (root) node tree is 1. Most of the recursive algorithms that a websearch returns, do use this.
2. Height of one (root) node tree is 0. Wikipedia (and its sources) do use this. Height of the tree is the height of the root node and the height of a node is the number of edges on longest simple path from it to a leaf. While the definition is explicit, there are practically no code examples.