I cannot understand the first paragraph of the proof, which comes from the known book Introduction to Algorithms, third-edition, and I consider it has some errors, could anyone help me check about it?
Possible errors:
It first prove the case $\text{height(x)=0},$ then it says "For inductive steps, consider a node $x$ that has positive height".
From my understanding of inductive proof, the base case should be able to trigger the inductive statement. I mean: the "first domino" should trigger the next one, so the statement should be something like non-negative height.
It says "each child has a black-height of either $\text{bh}(x)$ or $\text{bh}(x)-1$", but when applying, only the latter is used: $(2^{\text{bh}(x)-1}-1)+(2^{\text{bh}(x)-1}-1)+1=2^{\text{bh}(x)}-1$.